diff --git a/Doc/pages/R_notation.rst b/Doc/pages/R_notation.rst index 8d0046d64..22f8e482b 100644 --- a/Doc/pages/R_notation.rst +++ b/Doc/pages/R_notation.rst @@ -4,41 +4,46 @@ Frequently Used Symbols and Notations ===================================== -============================================ =============== -**Symbol** **Description** -:math:`\alpha, \beta, \gamma, \ldots` atom types -:math:`j, k, l, \ldots` atom indices -:math:`c_{\alpha}` concentration of atom-type :math:`\alpha` where :math:`c_{\alpha} = N_{\alpha} / N` -:math:`D` diffusion constant -:math:`F(\mathbf{q}, t)` intermediate scattering function -:math:`F_{\mathrm{coh}}(\mathbf{q}, t)` coherent intermediate scattering function -:math:`F_{\mathrm{inc}}(\mathbf{q}, t)` incoherent intermediate scattering function -:math:`g(\mathbf{r})` pair distribution function -:math:`G(\mathbf{r}, t)` the van Hove function -:math:`G_{\mathrm{d}}(\mathbf{r}, t)` distinct part of the van Hove function -:math:`G_{\mathrm{s}}(\mathbf{r}, t)` self part of the van Hove function -:math:`t` time -:math:`t_{\mathrm{cor}}` the total time of the correlation function -:math:`t_{\mathrm{tot}}` the total time of the MD simulation -:math:`\Delta t` the MD time step -:math:`S(\mathbf{q})` static structure factor -:math:`S(\mathbf{q}, \omega)` dynamic structure factor -:math:`S_{\mathrm{coh}}(\mathbf{q}, \omega)` dynamic coherent structure factor -:math:`S_{\mathrm{inc}}(\mathbf{q}, \omega)` dynamic incoherent structure factor -:math:`T` temperature -:math:`M` number of atom types in the system -:math:`n_{\mathrm{c}}` total number of time steps calculated for the correlation function -:math:`n_{\mathrm{t}}` total number of time steps in the MD simulation -:math:`N` total number of atoms in the unit cell, simulation or volume :math:`V` -:math:`N_\alpha` number of atom of type :math:`\alpha` -:math:`\rho` density -:math:`r` distance -:math:`q` wavenumber -:math:`\mathbf{q}` wavevector -:math:`\mathbf{r}(t)` the position of an atom at time :math:`t` -:math:`\mathbf{v}(t)` the velocity of an atom at time :math:`t` -:math:`W_\alpha` weighting factor for atom-type :math:`\alpha` -:math:`W_\alpha\beta` weighting factor for atom-type pair :math:`\alpha\beta` -:math:`\omega` angular frequency -:math:`\Delta \omega` the frequency step -============================================ =============== +======================================================= =============== +**Symbol** **Description** +:math:`\alpha, \beta, \gamma, \ldots` atom types +:math:`j, k, l, \ldots` atom indices +:math:`c_{\alpha}` concentration of atom-type :math:`\alpha` where :math:`c_{\alpha} = N_{\alpha} / N` +:math:`C_{\mathbf{vv}jj}(t)` the velocity autocorrelation function of particle :math:`j` +:math:`C_{\mathbf{vv}jj}(\omega)` the Fourier transform of the velocity autocorrelation function of particle :math:`j` +:math:`D` diffusion constant +:math:`\mathrm{DOS}(\omega)` density of states +:math:`F(\mathbf{q}, t)` intermediate scattering function +:math:`F_{\mathrm{coh}}(\mathbf{q}, t)` coherent intermediate scattering function +:math:`F_{\mathrm{inc}}(\mathbf{q}, t)` incoherent intermediate scattering function +:math:`F_{\mathrm{inc}}^{\text{G}}(\mathbf{q}, t)` Gaussian approximation of the incoherent intermediate scattering function +:math:`g(\mathbf{r})` pair distribution function +:math:`G(\mathbf{r}, t)` the van Hove function +:math:`G_{\mathrm{d}}(\mathbf{r}, t)` distinct part of the van Hove function +:math:`G_{\mathrm{s}}(\mathbf{r}, t)` self part of the van Hove function +:math:`t` time +:math:`t_{\mathrm{cor}}` the total time of the correlation function +:math:`t_{\mathrm{tot}}` the total time of the MD simulation +:math:`\Delta t` the MD time step +:math:`S(\mathbf{q})` static structure factor +:math:`S(\mathbf{q}, \omega)` dynamic structure factor +:math:`S_{\mathrm{coh}}(\mathbf{q}, \omega)` dynamic coherent structure factor +:math:`S_{\mathrm{inc}}(\mathbf{q}, \omega)` dynamic incoherent structure factor +:math:`S_{\mathrm{inc}}^{\text{G}}(\mathbf{q}, \omega)` Gaussian approximation of the dynamic incoherent structure factor +:math:`T` temperature +:math:`M` number of atom types in the system +:math:`n_{\mathrm{c}}` total number of time steps calculated for the correlation function +:math:`n_{\mathrm{t}}` total number of time steps in the MD simulation +:math:`N` total number of atoms in the unit cell, simulation or volume :math:`V` +:math:`N_\alpha` number of atom of type :math:`\alpha` +:math:`\rho` density +:math:`r` distance +:math:`q` wavenumber +:math:`\mathbf{q}` wavevector +:math:`\mathbf{r}(t)` the position of an atom at time :math:`t` +:math:`\mathbf{v}(t)` the velocity of an atom at time :math:`t` +:math:`W_\alpha` weighting factor for atom-type :math:`\alpha` +:math:`W_\alpha\beta` weighting factor for atom-type pair :math:`\alpha\beta` +:math:`\omega` angular frequency +:math:`\Delta \omega` the frequency step +======================================================= =============== diff --git a/Doc/pages/analysis.rst b/Doc/pages/analysis.rst index 90505e616..15e61d8bd 100644 --- a/Doc/pages/analysis.rst +++ b/Doc/pages/analysis.rst @@ -102,292 +102,328 @@ Trajectory Box Translated Trajectory ''''''''''''''''''''''''' -A box translated trajectory in molecular dynamics simulations refers to a -technique where the entire simulation box, representing the space in which -molecules interact, is shifted or translated during the simulation. This -approach can be useful for correcting periodic boundary condition artifacts, -studying different regions of a system, applying unique boundary conditions, -or mitigating surface effects. The translation of the simulation box allows -researchers to explore specific aspects of molecular behavior and system -properties within the computational environment. +.. note:: + + **This job is under development MDANSE and is currently not available. + The documentation here is out-dated and only left here for referencing + purposes.** + + A box translated trajectory in molecular dynamics simulations refers to a + technique where the entire simulation box, representing the space in which + molecules interact, is shifted or translated during the simulation. This + approach can be useful for correcting periodic boundary condition artifacts, + studying different regions of a system, applying unique boundary conditions, + or mitigating surface effects. The translation of the simulation box allows + researchers to explore specific aspects of molecular behavior and system + properties within the computational environment. .. _center-of-masses-trajectory: Center Of Masses Trajectory ''''''''''''''''''''''''''' -The center of mass trajectory (COMT) analysis consists in deriving the -trajectory of the respective centres of mass of a set of groups of -atoms. In order to produce a visualizable trajectory, MDANSE assigns -the centres of mass to pseudo-hydrogen atoms whose mass is equal to the -mass of their associated group. Thus, the produced trajectory can be -reused for other analysis. In that sense, COMT analysis is a practical -way to reduce noticeably the dimensionality of a system. +.. note:: + + **This job is under development MDANSE and is currently not available. + The documentation here is out-dated and only left here for referencing + purposes.** + + The center of mass trajectory (COMT) analysis consists in deriving the + trajectory of the respective centres of mass of a set of groups of + atoms. In order to produce a visualizable trajectory, MDANSE assigns + the centres of mass to pseudo-hydrogen atoms whose mass is equal to the + mass of their associated group. Thus, the produced trajectory can be + reused for other analysis. In that sense, COMT analysis is a practical + way to reduce noticeably the dimensionality of a system. .. _cropped-trajectory: Cropped Trajectory '''''''''''''''''' -A cropped trajectory in molecular dynamics simulations refers to a -shortened version of the trajectory data file, focusing on a specific time -segment of a simulation. This cropping process is useful for reducing data -size, isolating relevant events, improving computational efficiency, and -enhancing visualization. It allows researchers to concentrate on the critical -dynamics or interactions within a molecular system while excluding -unnecessary or transient data. +.. note:: + + **This job is under development MDANSE and is currently not available. + The documentation here is out-dated and only left here for referencing + purposes.** + + A cropped trajectory in molecular dynamics simulations refers to a + shortened version of the trajectory data file, focusing on a specific time + segment of a simulation. This cropping process is useful for reducing data + size, isolating relevant events, improving computational efficiency, and + enhancing visualization. It allows researchers to concentrate on the critical + dynamics or interactions within a molecular system while excluding + unnecessary or transient data. .. _global-motion-filtered-trajectory: Global Motion Filtered Trajectory ''''''''''''''''''''''''''''''''' -It is often of interest to separate global motion from internal motion, -both for quantitative analysis and for visualization by animated -display. Obviously, this can be done under the hypothesis that global -and internal motions are decoupled within the length and timescales of -the analysis. MDANSE can create global motion filtered trajectory -(GMFT) by filtering out global motions (made of the three -translational and rotational degrees of freedom), either on the whole -system or on a user-defined subset, by fitting it to a reference -structure (usually the first frame of the MD). Global motion filtering -uses a straightforward algorithm: - -- for the first frame, find the linear transformation such that the - coordinate origin becomes the centre of mass of the system and its - principal axes of inertia are parallel to the three coordinates axes - (also called principal axes transformation), -- this provides a reference configuration :math:`C_{\mathrm{ref}}`, -- for any other frames :math:`f`, finds and applies the linear transformation - that minimizes the RMS distance between frame :math:`f` and :math:`C_{\mathrm{ref}}`. - -The result is stored in a new trajectory file that contains only -internal motions. This analysis can be useful in case where diffusive -motions are not of interest or simply not accessible to the experiment -(time resolution, powder analysis . . . ). +.. note:: + + **This job is under development MDANSE and is currently not available. + The documentation here is out-dated and only left here for referencing + purposes.** + + It is often of interest to separate global motion from internal motion, + both for quantitative analysis and for visualization by animated + display. Obviously, this can be done under the hypothesis that global + and internal motions are decoupled within the length and timescales of + the analysis. MDANSE can create global motion filtered trajectory + (GMFT) by filtering out global motions (made of the three + translational and rotational degrees of freedom), either on the whole + system or on a user-defined subset, by fitting it to a reference + structure (usually the first frame of the MD). Global motion filtering + uses a straightforward algorithm: + + - for the first frame, find the linear transformation such that the + coordinate origin becomes the centre of mass of the system and its + principal axes of inertia are parallel to the three coordinates axes + (also called principal axes transformation), + - this provides a reference configuration :math:`C_{\mathrm{ref}}`, + - for any other frames :math:`f`, finds and applies the linear transformation + that minimizes the RMS distance between frame :math:`f` and :math:`C_{\mathrm{ref}}`. + + The result is stored in a new trajectory file that contains only + internal motions. This analysis can be useful in case where diffusive + motions are not of interest or simply not accessible to the experiment + (time resolution, powder analysis . . . ). .. _rigid-body-trajectory: Rigid Body Trajectory ''''''''''''''''''''' -To analyse the dynamics of complex molecular systems it is often -desirable to consider the overall motion of molecules or molecular -subunits. We will call this motion rigid-body motion in the following. -Rigid-body motions are fully determined by the dynamics of the centroid, -which may be the centre-of-mass, and the dynamics of the angular -coordinates describing the orientation of the rigid body. The angular -coordinates are the appropriate variables to compute angular correlation -functions of molecular systems in space and time. In most cases, -however, these variables are not directly available from MD -simulations since MD algorithms typically work in cartesian -coordinates. Molecules are either treated as flexible, or, if they are -treated as rigid, constraints are taken into account in the framework of -cartesian coordinates [Ref23]_. In MDANSE, -rigid-body trajectory (RBT) can be defined from a MD trajectory by -fitting rigid reference structures, defining a (sub)molecule, to the -corresponding structure in each time frame of the trajectory. Here 'fit' -means the optimal superposition of the structures in a least-squares -sense. We will describe now how rigid body motions, i.e. global -translations and rotations of molecules or subunits of complex -molecules, can be extracted from a MD trajectory. A more detailed -presentation is given in [Ref24]_. We define -an optimal rigid-body trajectory in the following way: for each time -frame of the trajectory the atomic positions of a rigid reference -structure, defined by the three cartesian components of its centroid -(e.g. the centre of mass) and three angles, are as close as possible to -the atomic positions of the corresponding structure in the MD -configuration. Here "as close as possible" means as close as possible in -a least-squares sense. - -**Optimal superposition**: We consider a given time frame in which the -atomic positions of a (sub)molecule are given by :math:`x_{\alpha}` where :math:`{\alpha = 1}, \ldots, N`. -The corresponding positions in the reference structure are denoted as -:math:`x_{\alpha}^{(0)}` where :math:`{\alpha = 1}, \ldots, N`. -For both the given structure and the reference structure we introduce -the yet undetermined centroids :math:`X` and :math:`X^{(0)}`, respectively, and -define the deviation +.. note:: + + **This job is under development MDANSE and is currently not available. + The documentation here is out-dated and only left here for referencing + purposes.** + + To analyse the dynamics of complex molecular systems it is often + desirable to consider the overall motion of molecules or molecular + subunits. We will call this motion rigid-body motion in the following. + Rigid-body motions are fully determined by the dynamics of the centroid, + which may be the centre-of-mass, and the dynamics of the angular + coordinates describing the orientation of the rigid body. The angular + coordinates are the appropriate variables to compute angular correlation + functions of molecular systems in space and time. In most cases, + however, these variables are not directly available from MD + simulations since MD algorithms typically work in cartesian + coordinates. Molecules are either treated as flexible, or, if they are + treated as rigid, constraints are taken into account in the framework of + cartesian coordinates [Ref23]_. In MDANSE, + rigid-body trajectory (RBT) can be defined from a MD trajectory by + fitting rigid reference structures, defining a (sub)molecule, to the + corresponding structure in each time frame of the trajectory. Here 'fit' + means the optimal superposition of the structures in a least-squares + sense. We will describe now how rigid body motions, i.e. global + translations and rotations of molecules or subunits of complex + molecules, can be extracted from a MD trajectory. A more detailed + presentation is given in [Ref24]_. We define + an optimal rigid-body trajectory in the following way: for each time + frame of the trajectory the atomic positions of a rigid reference + structure, defined by the three cartesian components of its centroid + (e.g. the centre of mass) and three angles, are as close as possible to + the atomic positions of the corresponding structure in the MD + configuration. Here "as close as possible" means as close as possible in + a least-squares sense. + + **Optimal superposition**: We consider a given time frame in which the + atomic positions of a (sub)molecule are given by :math:`x_{\alpha}` where :math:`{\alpha = 1}, \ldots, N`. + The corresponding positions in the reference structure are denoted as + :math:`x_{\alpha}^{(0)}` where :math:`{\alpha = 1}, \ldots, N`. + For both the given structure and the reference structure we introduce + the yet undetermined centroids :math:`X` and :math:`X^{(0)}`, respectively, and + define the deviation -.. math:: - :label: pfx147 + .. math:: + :label: pfx147 - {\Delta_{\alpha}\doteq D(q){\left\lbrack {x_{\alpha}^{(0)} - X^{(0)}} \right\rbrack - \left\lbrack {x_{\alpha} - X} \right\rbrack}.} + {\Delta_{\alpha}\doteq D(q){\left\lbrack {x_{\alpha}^{(0)} - X^{(0)}} \right\rbrack - \left\lbrack {x_{\alpha} - X} \right\rbrack}.} -Here :math:`D(q)` is a rotation matrix which depends on also yet -undetermined angular coordinates which we chose to be quaternion -parameters, abbreviated as vector :math:`q = (q_0, q_1, q_2, q_3)`. -The quaternion parameters fulfil the normalization condition :math:`q \cdot {q = 1}` [Ref25]_. -The target function to be minimized is now defined as + Here :math:`D(q)` is a rotation matrix which depends on also yet + undetermined angular coordinates which we chose to be quaternion + parameters, abbreviated as vector :math:`q = (q_0, q_1, q_2, q_3)`. + The quaternion parameters fulfil the normalization condition :math:`q \cdot {q = 1}` [Ref25]_. + The target function to be minimized is now defined as -.. math:: - :label: pfx149 + .. math:: + :label: pfx149 - {m{\left( {q;X,X^{(0)}} \right) = {\sum\limits_{\alpha}{\omega_{\alpha}|\Delta|_{\alpha}^{2}}}}.} + {m{\left( {q;X,X^{(0)}} \right) = {\sum\limits_{\alpha}{\omega_{\alpha}|\Delta|_{\alpha}^{2}}}}.} -where :math:`\omega_{\alpha}` are atomic weights. The minimization -with respect to the centroids is decoupled from the minimization with -respect to the quaternion parameters and yields + where :math:`\omega_{\alpha}` are atomic weights. The minimization + with respect to the centroids is decoupled from the minimization with + respect to the quaternion parameters and yields -.. math:: - :label: pfx150 + .. math:: + :label: pfx150 - {{X = {\sum\limits_{\alpha}\omega_{\alpha}}}x_{\alpha} \qquad\qquad {X^{(0)} = {\sum\limits_{\alpha}\omega_{\alpha}}}x_{\alpha}^{(0)}} + {{X = {\sum\limits_{\alpha}\omega_{\alpha}}}x_{\alpha} \qquad\qquad {X^{(0)} = {\sum\limits_{\alpha}\omega_{\alpha}}}x_{\alpha}^{(0)}} -We are now left with a minimization problem for the rotational part -which can be written as + We are now left with a minimization problem for the rotational part + which can be written as -.. math:: - :label: pfx152 + .. math:: + :label: pfx152 - m{(q) = {\sum\limits_{\alpha}{\omega_{\alpha}\left\lbrack {{D(q)r}_{\alpha}^{(0)} - r_{\alpha}} \right\rbrack^{2}}}\overset{!}{=}\mathrm{Min}}. + m{(q) = {\sum\limits_{\alpha}{\omega_{\alpha}\left\lbrack {{D(q)r}_{\alpha}^{(0)} - r_{\alpha}} \right\rbrack^{2}}}\overset{!}{=}\mathrm{Min}}. -The relative position vectors + The relative position vectors -.. math:: - :label: pfx153 + .. math:: + :label: pfx153 - {{r_{\alpha} = {x_{\alpha} - X}} \qquad\qquad r_{\alpha}^{(0)} = {x_{\alpha}^{(0)} - X^{(0)}}} + {{r_{\alpha} = {x_{\alpha} - X}} \qquad\qquad r_{\alpha}^{(0)} = {x_{\alpha}^{(0)} - X^{(0)}}} -are fixed and the rotation matrix reads -[Ref25]_ + are fixed and the rotation matrix reads + [Ref25]_ -.. math:: - :label: pfx155 + .. math:: + :label: pfx155 - D(q) = \begin{pmatrix} - {q_{0}^{2} + q_{1}^{2} - q_{2}^{2} - q_{3}^{2}} & {2\left( {{- q_{0}}{q_{3} + q_{1}}q_{2}} \right)} & {2\left( {q_{0}{q_{2} + q_{1}}q_{3}} \right)} \\ - {2\left( {q_{0}{q_{3} + q_{1}}q_{2}} \right)} & {q_{0}^{2} + q_{2}^{2} - q_{1}^{2} - q_{3}^{2}} & {2\left( {{- q_{0}}{q_{1} + q_{2}}q_{3}} \right)} \\ - {2\left( {{- q_{0}}{q_{2} + q_{1}}q_{3}} \right)} & {2\left( {q_{0}{q_{1} + q_{2}}q_{3}} \right)} & {q_{0}^{2} + q_{3}^{2} - q_{1}^{2} - q_{2}^{2}} \\ - \end{pmatrix} + D(q) = \begin{pmatrix} + {q_{0}^{2} + q_{1}^{2} - q_{2}^{2} - q_{3}^{2}} & {2\left( {{- q_{0}}{q_{3} + q_{1}}q_{2}} \right)} & {2\left( {q_{0}{q_{2} + q_{1}}q_{3}} \right)} \\ + {2\left( {q_{0}{q_{3} + q_{1}}q_{2}} \right)} & {q_{0}^{2} + q_{2}^{2} - q_{1}^{2} - q_{3}^{2}} & {2\left( {{- q_{0}}{q_{1} + q_{2}}q_{3}} \right)} \\ + {2\left( {{- q_{0}}{q_{2} + q_{1}}q_{3}} \right)} & {2\left( {q_{0}{q_{1} + q_{2}}q_{3}} \right)} & {q_{0}^{2} + q_{3}^{2} - q_{1}^{2} - q_{2}^{2}} \\ + \end{pmatrix} -**Quaternions and rotations**: The rotational minimization problem can -be elegantly solved by using quaternion algebra. Quaternions are -so-called hypercomplex numbers, having a real unit, 1, and three -imaginary units, :math:`I`, :math:`J`, and :math:`K`. Since :math:`IJ = K` (cyclic), -quaternion multiplication is not commutative. A possible matrix -representation of an arbitrary quaternion, + **Quaternions and rotations**: The rotational minimization problem can + be elegantly solved by using quaternion algebra. Quaternions are + so-called hypercomplex numbers, having a real unit, 1, and three + imaginary units, :math:`I`, :math:`J`, and :math:`K`. Since :math:`IJ = K` (cyclic), + quaternion multiplication is not commutative. A possible matrix + representation of an arbitrary quaternion, -.. math:: - :label: pfx156 + .. math:: + :label: pfx156 - {{A = a_{0}}{1 + a_{1}}{I + a_{2}}{J + a_{3}} K,} + {{A = a_{0}}{1 + a_{1}}{I + a_{2}}{J + a_{3}} K,} -reads + reads -.. math:: - :label: pfx157 + .. math:: + :label: pfx157 - A = \begin{pmatrix} - a_{0} & {- a_{1}} & {- a_{2}} & {- a_{3}} \\ - a_{1} & a_{0} & {- a_{3}} & a_{2} \\ - a_{2} & a_{3} & a_{0} & {- a_{1}} \\ - a_{3} & {- a_{2}} & a_{1} & a_{0} \\ - \end{pmatrix} + A = \begin{pmatrix} + a_{0} & {- a_{1}} & {- a_{2}} & {- a_{3}} \\ + a_{1} & a_{0} & {- a_{3}} & a_{2} \\ + a_{2} & a_{3} & a_{0} & {- a_{1}} \\ + a_{3} & {- a_{2}} & a_{1} & a_{0} \\ + \end{pmatrix} -The components :math:`a_{\upsilon}` -are real numbers. Similarly, as normal complex numbers allow one to -represent rotations in a plane, quaternions allow one to represent -rotations in space. Consider the quaternion representation of a vector -:math:`R`, which is given by + The components :math:`a_{\upsilon}` + are real numbers. Similarly, as normal complex numbers allow one to + represent rotations in a plane, quaternions allow one to represent + rotations in space. Consider the quaternion representation of a vector + :math:`R`, which is given by -.. math:: - :label: pfx158 + .. math:: + :label: pfx158 - {{R = x}{I + y}{J + z} K,} + {{R = x}{I + y}{J + z} K,} -and perform the operation + and perform the operation -.. math:: - :label: pfx159 + .. math:: + :label: pfx159 - {{R^{'} = \mathit{QRQ}^{T}},} + {{R^{'} = \mathit{QRQ}^{T}},} -where :math:`Q` is a normalised quaternion + where :math:`Q` is a normalised quaternion -.. math:: - :label: pfx160 + .. math:: + :label: pfx160 - {\text{|}Q\text{|}^{2}\doteq{{q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2}} = \frac{1}{4}\mathrm{Tr}\, Q^{T}Q = 1}}. + {\text{|}Q\text{|}^{2}\doteq{{q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2}} = \frac{1}{4}\mathrm{Tr}\, Q^{T}Q = 1}}. -We note that a normalized quaternion is represented by an orthogonal 4 x 4 matrix. :math:`R'` may then be -written as + We note that a normalized quaternion is represented by an orthogonal 4 x 4 matrix. :math:`R'` may then be + written as -.. math:: - :label: pfx161 + .. math:: + :label: pfx161 - {{R^{'} = x^{'}}{I + y^{'}}{J + z^{'}} K,} + {{R^{'} = x^{'}}{I + y^{'}}{J + z^{'}} K,} -where the components :math:`x'`, :math:`y'`, :math:`z'`, abbreviated as :math:`r'`, are given by :math:`r^{'} = D(q)r`. + where the components :math:`x'`, :math:`y'`, :math:`z'`, abbreviated as :math:`r'`, are given by :math:`r^{'} = D(q)r`. -**Solution of the minimization problem**: In quaternion algebra, the -rotational minimization problem may now be phrased as follows: + **Solution of the minimization problem**: In quaternion algebra, the + rotational minimization problem may now be phrased as follows: -.. math:: - :label: pfx163 + .. math:: + :label: pfx163 - {m{(q) = {{\sum\limits_{\alpha}{{\omega_{\alpha}\text{|}\mathit{QR}}_{\alpha}^{(0)}Q}^{T}} - R_{\alpha}}}{\text{|}^{2}\overset{!}{=}\mathrm{Min}}.} + {m{(q) = {{\sum\limits_{\alpha}{{\omega_{\alpha}\text{|}\mathit{QR}}_{\alpha}^{(0)}Q}^{T}} - R_{\alpha}}}{\text{|}^{2}\overset{!}{=}\mathrm{Min}}.} -Since the matrix :math:`Q` representing a normalized quaternion is orthogonal -this may also be written as + Since the matrix :math:`Q` representing a normalized quaternion is orthogonal + this may also be written as -.. math:: - :label: pfx164 + .. math:: + :label: pfx164 - {{{m{(q) = {\sum\limits_{\alpha}\omega_{\alpha}}}\text{|}\mathit{QR}_{\alpha}^{(0)}} - R_{\alpha}}Q\text{|}^{2}{\overset{!}{=}\mathrm{Min}}.} + {{{m{(q) = {\sum\limits_{\alpha}\omega_{\alpha}}}\text{|}\mathit{QR}_{\alpha}^{(0)}} - R_{\alpha}}Q\text{|}^{2}{\overset{!}{=}\mathrm{Min}}.} -This follows from the simple fact that :math:`\text{|}A{\text{|} = \text{|}}\mathit{AQ}\text{|}` -if :math:`Q` is normalized. Eq. `104` shows that the -target function to be minimized can be written as a simple quadratic -form in the quaternion parameters [Ref24]_, + This follows from the simple fact that :math:`\text{|}A{\text{|} = \text{|}}\mathit{AQ}\text{|}` + if :math:`Q` is normalized. Eq. `104` shows that the + target function to be minimized can be written as a simple quadratic + form in the quaternion parameters [Ref24]_, -.. math:: - :label: pfx166 + .. math:: + :label: pfx166 - {m{(q) = q}\cdot\mathit{Mq} \qquad\qquad {M = {\sum\limits_{\alpha}{\omega_{\alpha}M_{\alpha}}}}} + {m{(q) = q}\cdot\mathit{Mq} \qquad\qquad {M = {\sum\limits_{\alpha}{\omega_{\alpha}M_{\alpha}}}}} -The matrices :math:`M` are positive semi-definite matrices depending on the -positions :math:`r_{\alpha}` and :math:`r_{\alpha}^{(0)}`. + The matrices :math:`M` are positive semi-definite matrices depending on the + positions :math:`r_{\alpha}` and :math:`r_{\alpha}^{(0)}`. -The rotational fit is now reduced to the problem of finding the minimum -of a quadratic form with the constraint that the quaternion to be -determined must be normalized. Using the method of Lagrange multipliers -to account for the normalization constraint we have + The rotational fit is now reduced to the problem of finding the minimum + of a quadratic form with the constraint that the quaternion to be + determined must be normalized. Using the method of Lagrange multipliers + to account for the normalization constraint we have -.. math:: - :label: pfx169 + .. math:: + :label: pfx169 - {m^{'}{\left( {q,\lambda} \right) = q}\cdot{\mathit{Mq} - \lambda}{\left( {q\cdot{q - 1}} \right)\overset{!}{=}\mathrm{Min}}.} + {m^{'}{\left( {q,\lambda} \right) = q}\cdot{\mathit{Mq} - \lambda}{\left( {q\cdot{q - 1}} \right)\overset{!}{=}\mathrm{Min}}.} -This leads immediately to the eigenvalue problem + This leads immediately to the eigenvalue problem -.. math:: - :label: pfx170 + .. math:: + :label: pfx170 - {{\mathit{Mq} = \lambda}q \qquad\qquad q\cdot{q = 1.}} + {{\mathit{Mq} = \lambda}q \qquad\qquad q\cdot{q = 1.}} -Now any normalized eigenvector :math:`q` fulfils the relation + Now any normalized eigenvector :math:`q` fulfils the relation -.. math:: - :label: pfx172 - - {{\lambda = q}\cdot\mathit{Mq}\equiv m(q)} + .. math:: + :label: pfx172 -Therefore, the eigenvector belonging to the smallest eigenvalue, -:math:`\lambda_{\mathrm{min}}`, is the desired solution. At the same time :math:`\lambda_{\mathrm{min}}` -gives the average error per atom. The result of RBT analysis is stored -in a new trajectory file that contains only RBT motions. + {{\lambda = q}\cdot\mathit{Mq}\equiv m(q)} + + Therefore, the eigenvector belonging to the smallest eigenvalue, + :math:`\lambda_{\mathrm{min}}`, is the desired solution. At the same time :math:`\lambda_{\mathrm{min}}` + gives the average error per atom. The result of RBT analysis is stored + in a new trajectory file that contains only RBT motions. .. _unfolded-trajectory: Unfolded Trajectory ''''''''''''''''''' -An unfolded trajectory in the context of molecular dynamics -simulations refers to a trajectory data file that has been processed or -analyzed to reveal the unfolding or expansion of molecular structures over -time. This term is particularly relevant in the study of biomolecules or -polymers, where understanding the dynamic evolution and changes in these -structures holds significant importance for scientific applications, -including drug design, materials science, and biomolecular research. -Unfolding trajectories provide valuable insights into molecular behavior -and interactions, contributing to the development of new materials and the -design of therapeutic compounds. +.. note:: + + **This job is under development MDANSE and is currently not available. + The documentation here is out-dated and only left here for referencing + purposes.** + + An unfolded trajectory in the context of molecular dynamics + simulations refers to a trajectory data file that has been processed or + analyzed to reveal the unfolding or expansion of molecular structures over + time. This term is particularly relevant in the study of biomolecules or + polymers, where understanding the dynamic evolution and changes in these + structures holds significant importance for scientific applications, + including drug design, materials science, and biomolecular research. + Unfolding trajectories provide valuable insights into molecular behavior + and interactions, contributing to the development of new materials and the + design of therapeutic compounds. Virtual Instruments @@ -399,11 +435,17 @@ Virtual Instruments McStas Virtual Instrument ''''''''''''''''''''''''' -McStas enables researchers to create virtual instruments that replicate the -behavior of real neutron or X-ray instruments. This capability streamlines -the design, optimization, and testing of experiments within a virtual -environment before conducting physical experiments. Such simulations help -researchers conserve valuable time and resources while simultaneously -enhancing the precision and reliability of their experiments. McStas finds -widespread application in fields like materials science and condensed -matter physics. +.. note:: + + **This job is under development MDANSE and is currently not available. + The documentation here is out-dated and only left here for referencing + purposes.** + + McStas enables researchers to create virtual instruments that replicate the + behavior of real neutron or X-ray instruments. This capability streamlines + the design, optimization, and testing of experiments within a virtual + environment before conducting physical experiments. Such simulations help + researchers conserve valuable time and resources while simultaneously + enhancing the precision and reliability of their experiments. McStas finds + widespread application in fields like materials science and condensed + matter physics. diff --git a/Doc/pages/correlation.rst b/Doc/pages/correlation.rst index 75532d197..2f56ef20a 100644 --- a/Doc/pages/correlation.rst +++ b/Doc/pages/correlation.rst @@ -15,24 +15,24 @@ is done in a consistent way. Consider two time series .. math:: :label: eqn-fca1 - A(k \Delta t) \quad \text{and} \quad B(k \Delta t) \qquad k = 0, \ldots, n_{\mathrm{t}}-1, + A(n \Delta t) \quad \text{and} \quad B(n \Delta t) \qquad n = 0, \ldots, n_{\mathrm{t}}-1, of length :math:`t_{\mathrm{tot}} = (n_{\mathrm{t}} -1) \Delta t` which are to be correlated. In MDANSE, correlation function are calculated by first choosing a specific number of correlation time steps :math:`n_{\mathrm{c}}` which will define the length of our correlation function :math:`t_{\mathrm{cor}} = (n_{\mathrm{c}} -1) \Delta t`. The correlation function of -:math:`A(k \Delta t)` and :math:`B(k \Delta t)` will be +:math:`A(n \Delta t)` and :math:`B(n \Delta t)` will be .. math:: :label: eqn-fca2 - C_{AB}(l \Delta t) = \frac{1}{n_{\mathrm{t}} - n_{\mathrm{c}} + 1} \sum\limits_{k=0}^{n_{\mathrm{t}} - n_{\mathrm{c}} + 1} A^{*}(k\Delta t)B([k + l]\Delta t) \qquad l = 0, \ldots, n_{c} - 1. + C_{AB}(n' \Delta t) = \frac{1}{n_{\mathrm{t}} - n_{\mathrm{c}} + 1} \sum\limits_{n=0}^{n_{\mathrm{t}} - n_{\mathrm{c}} + 1} A^{*}(n\Delta t)B([n + n']\Delta t) \qquad n' = 0, \ldots, n_{c} - 1. -In case that :math:`A(k \Delta t)` and -:math:`B(k \Delta t)` are identical, the corresponding correlation function -:math:`C_{AA}(l \Delta t)` is called an *autocorrelation* function. Notice that -the prefactor is the same for all :math:`l \Delta t` time steps, this was +In case that :math:`A(n \Delta t)` and +:math:`B(n \Delta t)` are identical, the corresponding correlation function +:math:`C_{AA}(n' \Delta t)` is called an *autocorrelation* function. Notice that +the prefactor is the same for all :math:`n' \Delta t` time steps, this was not the case in previous versions of MDANSE. This meant that for different time steps a different number of configurations were used to obtain the average correlation; leading to spuriously large correlations for some @@ -48,14 +48,14 @@ MDANSE the spectra can be smoothed by applying an instrument resolution function .. math:: - :label: eqn-fca11 + :label: eqn-fca3 - P_{AB}\left(m \Delta \omega \right) = \frac{\Delta t}{2 \pi}\sum_{l=-(n_{\mathrm{c}}-1)}^{n_{\mathrm{c}}-1} - \exp\left[- 2 \pi i \frac{l \Delta t }{2n_{\mathrm{c}} - 1} m \Delta \omega \right] \frac{W(l \Delta t)}{W(0)} C_{AB}( \vert l \Delta t \vert ) + P_{AB}\left(m \Delta \omega \right) = \frac{\Delta t}{2 \pi}\sum_{n=-(n_{\mathrm{c}}-1)}^{n_{\mathrm{c}}-1} + \exp\left[- 2 \pi i \frac{n \Delta t }{2n_{\mathrm{c}} - 1} m \Delta \omega \right] \frac{W(n \Delta t)}{W(0)} C_{AB}( \vert n \Delta t \vert ) here :math:`m = -(n_{\mathrm{c}}-1), \ldots, n_{\mathrm{c}}-1` and :math:`\Delta \omega` is the frequency step. Notice that the we assume that the correlation is symmetric so -that :math:`C_{AB}(l \Delta t) = C_{AB}( |l \Delta t| )` which should +that :math:`C_{AB}(n \Delta t) = C_{AB}( |n \Delta t| )` which should approximately be the case for all the correlation functions calculated in MDANSE assuming good (equilibrated, of a sufficient length/size and etc) MD trajectories are used. In MDANSE, the resolution function are @@ -63,11 +63,11 @@ specified in the frequency domain and are related to the resolution function in the time domain via a Fourier transform. .. math:: - :label: eqn-fca12 + :label: eqn-fca4 - W(l \Delta t) = \frac{1}{2n_{\mathrm{c}} - 1} \frac{1}{ \Delta t} \sum_{m=-(n_{\mathrm{c}}-1)}^{n_{\mathrm{c}}-1} \exp\left[ 2 \pi i \frac{m \Delta \omega}{2n_{\mathrm{c}} - 1} l \Delta t \right] W(m \Delta \omega) + W(n \Delta t) = \frac{1}{2n_{\mathrm{c}} - 1} \frac{1}{ \Delta t} \sum_{m=-(n_{\mathrm{c}}-1)}^{n_{\mathrm{c}}-1} \exp\left[ 2 \pi i \frac{m \Delta \omega}{2n_{\mathrm{c}} - 1} n \Delta t \right] W(m \Delta \omega) -where :math:`l = -(n_{\mathrm{c}}-1), \ldots, n_{\mathrm{c}}-1`. +where :math:`n = -(n_{\mathrm{c}}-1), \ldots, n_{\mathrm{c}}-1`. Resolution Functions ~~~~~~~~~~~~~~~~~~~~ @@ -81,7 +81,7 @@ are and whether longer MD trajectories where required. **Gaussian**: A Gaussian window instrument resolution function is .. math:: - :label: eqn-fca13 + :label: eqn-fca5 W(m \Delta \omega) = \frac{\sqrt{2 \pi}}{ \sigma} \exp\left[-\frac{1}{2}\left(\frac{ m \Delta \omega - \mu }{\sigma}\right)^2\right] @@ -91,7 +91,7 @@ and :math:`\mu` is a parameter which shifts the resolution function. **Lorentzian**: A Lorentzian window instrument resolution function is .. math:: - :label: eqn-fca14 + :label: eqn-fca6 W(m \Delta \omega) = \frac{2 \sigma}{(m\Delta \omega - \mu)^2 + \sigma^2} @@ -101,7 +101,7 @@ and :math:`\mu` is a parameter which shifts the resolution function.. **Triangular**: A triangular window instrument resolution function is .. math:: - :label: eqn-fca14 + :label: eqn-fca7 W(m \Delta \omega) = \begin{cases} 2 \pi (1 - \vert m \Delta \omega - \mu \vert / \sigma), & \vert m \Delta \omega - \mu \vert \leq \sigma;\\ @@ -114,7 +114,7 @@ and :math:`\mu` is a parameter which shifts the resolution function. **Square**: A square window instrument resolution function is .. math:: - :label: eqn-fca15 + :label: eqn-fca8 W(m \Delta \omega) = \begin{cases} \pi / \sigma, & \vert m \Delta \omega - \mu \vert \leq \sigma;\\ @@ -129,7 +129,7 @@ linear combination of the Gaussian and Lorentzian window functions .. math:: - :label: eqn-fca16 + :label: eqn-fca9 W(m \Delta \omega) = \eta \frac{2 \sigma_{\text{L}}}{(m\Delta \omega - \mu_{\text{L}})^2 + \sigma_{\text{L}}^2} + (1 - \eta) \frac{\sqrt{2 \pi}}{ \sigma_{\text{G}}} \exp\left[-\frac{1}{2}\left(\frac{ m \Delta \omega - \mu_{\text{G}} }{\sigma_{\text{G}}}\right)^2\right] diff --git a/Doc/pages/dynamics.rst b/Doc/pages/dynamics.rst index 9bc3fe7f7..4d6a438e1 100644 --- a/Doc/pages/dynamics.rst +++ b/Doc/pages/dynamics.rst @@ -20,38 +20,42 @@ This section contains background theory for following plugins: Angular Correlation ''''''''''''''''''' -The angular correlation analysis computes the autocorrelation of a set -of vectors describing the extent of a molecule in three orthogonal -directions. This kind of analysis can be useful when trying to highlight -the fact that a molecule is constrained in a given direction. +.. note:: -For a given triplet of non-colinear atoms :math:`g=(a_1,a_2,a_3)`, one can -derive an orthonormal set of three vectors :math:`v_1`, :math:`v_2`, :math:`v_3` using the -following scheme: + **This job is under development and may change in future versions + of MDANSE. The documentation here is out-dated and only left here + for referencing purposes.** -- :math:`v_{1} = (n_{1} + n_{2}) / \left| {n_{1} + n_{2}} \right|` - where :math:`n_1` and :math:`n_2` are respectively the - normalized vectors along (:math:`a_1`, :math:`a_2`) and (:math:`a_1`, :math:`a_3`) directions. -- :math:`v_2` is defined as the clockwise normal vector orthogonal to :math:`v_1` that - belongs to the plane defined by :math:`a_1`, :math:`a_2` and :math:`a_3` atoms -- :math:`v_{3} = v_{1}\times v_{2}` + The angular correlation analysis computes the autocorrelation of a set + of vectors describing the extent of a molecule in three orthogonal + directions. This kind of analysis can be useful when trying to highlight + the fact that a molecule is constrained in a given direction. -Thus, one can define the following autocorrelation functions for the -vectors :math:`v_1`, :math:`v_2` and :math:`v_3` defined on triplet :math:`i`: + For a given triplet of non-colinear atoms :math:`g=(a_1,a_2,a_3)`, one can + derive an orthonormal set of three vectors :math:`v_1`, :math:`v_2`, :math:`v_3` using the + following scheme: -.. math:: - :label: pfx3 + - :math:`v_{1} = (n_{1} + n_{2}) / \left| {n_{1} + n_{2}} \right|` + where :math:`n_1` and :math:`n_2` are respectively the + normalized vectors along (:math:`a_1`, :math:`a_2`) and (:math:`a_1`, :math:`a_3`) directions. + - :math:`v_2` is defined as the clockwise normal vector orthogonal to :math:`v_1` that + belongs to the plane defined by :math:`a_1`, :math:`a_2` and :math:`a_3` atoms + - :math:`v_{3} = v_{1}\times v_{2}` - {\mathrm{AC}_{g,i}{(t) = \left\langle {v_{g,i}(0)\cdot v_{g,i}(t)} \right\rangle} \qquad {i = 1,2,3}} + Thus, one can define the following autocorrelation functions for the + vectors :math:`v_1`, :math:`v_2` and :math:`v_3` defined on triplet :math:`i`: -and the angular correlation averaged over all triplets is: + .. math:: -.. math:: - :label: pfx4 + {\mathrm{AC}_{g,i}{(t) = \left\langle {v_{g,i}(0)\cdot v_{g,i}(t)} \right\rangle} \qquad {i = 1,2,3}} + + and the angular correlation averaged over all triplets is: - {\mathrm{AC}_{i}{(t) = {\sum\limits_{g = 1}^{N_{\mathrm{triplets}}}{\mathrm{AC}_{g,i}(t)}}} \qquad {i = 1,2,3}} + .. math:: -where :math:`N_{\mathrm{triplets}}` is the number of selected triplets. + {\mathrm{AC}_{i}{(t) = {\sum\limits_{g = 1}^{N_{\mathrm{triplets}}}{\mathrm{AC}_{g,i}(t)}}} \qquad {i = 1,2,3}} + + where :math:`N_{\mathrm{triplets}}` is the number of selected triplets. .. _analysis-dos: @@ -63,26 +67,29 @@ Density Of States MDANSE calculates the power spectrum of the VACF (see the section on :ref:`analysis-vacf`), which in case of -the mass-weighted VACF defines the phonon discrete DOS , defined as: +the mass-weighted VACF defines the phonon discrete DOS as: .. math:: - :label: pfx5 + :label: pfx1 - {\mathrm{DOS}( {\omega} ) = {\sum\limits_{\alpha}^{M} W_{\alpha}} {\widetilde{C}}_{\mathbf{vv};\alpha\alpha}\left(\omega \right)} + \mathrm{DOS}(\omega) = \sum\limits_{\alpha} W_{\alpha} C_{\mathbf{vv}\alpha\alpha}\left( \omega \right) +.. math:: + :label: pfx2 -where :math:`{\widetilde{C}}_{\mathbf{vv};\alpha\alpha}\left( \omega \right)` -is the Fourier transform of the velocity autocorrelation function average over atoms of type :math:`\alpha`, -:math:`W_{\alpha}` is the weighting factor of atom type :math:`\alpha` -and :math:`M` is the total number of atoms types in the system. + C_{\mathbf{vv}\alpha\alpha}(\omega) = \frac{1}{Nc_{\alpha}} \sum_{j}^{N_{\alpha}} \frac{1}{6\pi} \int\limits_{\infty}^{-\infty}\mathrm{d}t \, \left\langle \mathbf{v}_{j}\left( 0 \right)\cdot \mathbf{v}_{j}\left( t \right) \right\rangle e^{-i\omega t} +where :math:`C_{\mathbf{vv}\alpha\alpha}\left( \omega \right)` +is the Fourier transform of the velocity autocorrelation function average over atoms of type :math:`\alpha`, +:math:`W_{\alpha}` is the weighting factor of atom type :math:`\alpha`. The DOS can be computed either for the isotropic case or with respect to a -user-defined axis. Since the DOS -is computed from the unnormalized VACF, :math:`\mathrm{DOS}(0)` gives an +user-defined axis. + +Since the DOS is computed from the unnormalized VACF, the DOS at :math:`\omega=0` gives an approximate value for the diffusion constant (see Eqs. :math:numref:`pfx20`) when an equal weighting scheme is used. The DOS can be smoothed by, for example, a Gaussian window applied in the time domain -[Ref10]_ (see the section :ref:`appendix-fca`). We remark that the diffusion +[Ref10]_ (see the section :ref:`appendix-fca`); the diffusion constant obtained from this DOS is biased due to the spectral smoothing procedure since the VACF is weighted by this window Gaussian function. @@ -125,62 +132,62 @@ performed on a water box of 768 water molecules. To get the diffusion coefficient out of this plot, the slope of the linear part of the plot should be calculated. -By defining :math:`\mathbf{d}_{i}(t) = \mathbf{r}_{i}( t ) - \mathbf{r}_{i}( 0 )` -the MSD of particle :math:`i` can be written as: +By defining :math:`\mathbf{d}_{j}(t) = \mathbf{r}_{j}( t ) - \mathbf{r}_{j}( 0 )` +the MSD of particle :math:`j` can be written as: .. math:: - :label: pfx14 + :label: pfx3 - \Delta_{i}^{2}{(t) = \left\langle {d_{i}^{2}( {t} )} \right\rangle} + \Delta_{j}^{2}{(t) = \left\langle {d_{j}^{2}( {t} )} \right\rangle} -where :math:`\mathbf{r}_{i}(0)` and :math:`\mathbf{r}_{i}(t)` are -the position of particle :math:`i` at times :math:`0` and :math:`t` -and :math:`d_{i}( t ) = \vert \mathbf{d}_{i}(t) \vert`. +where :math:`\mathbf{r}_{j}(0)` and :math:`\mathbf{r}_{j}(t)` are +the position of particle :math:`j` at times :math:`0` and :math:`t` +and :math:`d_{j}( t ) = \vert \mathbf{d}_{j}(t) \vert`. One can introduce an MSD with respect to a given axis :math:`\mathbf{n}`: .. math:: - :label: pfx15 + :label: pfx4 - \Delta_{i}^{2}( t;\hat{\mathbf{n}} ) = \left\langle {d_{i}^{2}( {t;\hat{\mathbf{n}}} )} \right\rangle \qquad d_{i}(t;\hat{\mathbf{n}}) = \hat{\mathbf{n}} \cdot \mathbf{d}_{i}( t ) + \Delta_{j}^{2}( t, \hat{\mathbf{n}} ) = \left\langle {d_{j}^{2}( {t, \hat{\mathbf{n}}} )} \right\rangle \qquad d_{j}(t, \hat{\mathbf{n}}) = \hat{\mathbf{n}} \cdot \mathbf{d}_{j}( t ) where :math:`\hat{\mathbf{n}}` is a unit vector along :math:`\mathbf{n}`. -The calculation of MSD is the standard way to obtain diffusion -coefficients from Molecular Dynamics (MD) simulations. +The calculation of MSD is the standard way to obtain diffusion +coefficients from MD simulations. Assuming Einstein-diffusion in the long time limit one has for isotropic systems .. math:: - :label: pfx17 + :label: pfx5 - {D_{i} = {\lim\limits_{t\rightarrow\infty}{\frac{1}{6t}\mathrm{\Delta}_{i}^{2}(t)}}}. + {D_{j} = {\lim\limits_{t\rightarrow\infty}{\frac{1}{6t}\mathrm{\Delta}_{j}^{2}(t)}}}. There exists also a well-known relation between the MSD and the velocity autocorrelation function. One can show (see e.g. [Ref11]_) that .. math:: - :label: pfx18 + :label: pfx6 - \mathbf{d}_{i}{(t) = {\int\limits_{0}^{t}{\mathrm{d}t_1 \, \mathbf{v}_{i}(t_1)}}} \qquad \text{and} \qquad \mathrm{\Delta}_{i}^{2}{(t) = 6}{\int\limits_{0}^{t}{\mathrm{d}t_1 \, ( t - t_1 )C_{\mathbf{vv};ii}(t_1)}} + \mathbf{d}_{j}{(t) = {\int\limits_{0}^{t}{\mathrm{d}t' \, \mathbf{v}_{j}(t')}}} \qquad \text{and} \qquad \mathrm{\Delta}_{j}^{2}{(t) = 6}{\int\limits_{0}^{t}{\mathrm{d}t' \, ( t - t' )C_{\mathbf{vv}jj}(t')}} -where :math:`C_{\mathbf{vv};ii}(t)` is the velocity autocorrelation function of the particle :math:`i`. -Using now the definition Eq. :math:numref:`pfx17` of the diffusion +where :math:`C_{\mathbf{vv}jj}(t)` is the velocity autocorrelation function of the particle :math:`j`. +Using now the definition Eq. :math:numref:`pfx5` of the diffusion coefficient one obtains the relations .. math:: - :label: pfx20 + :label: pfx7 - {{D_{i} = {\int\limits_{0}^{\infty}{\mathrm{d}t \, C_{\mathbf{vv};ii}(t)}}} = \pi \widetilde{C}_{\mathbf{vv};ii}(0).} + {{D_{j} = {\int\limits_{0}^{\infty}{\mathrm{d}t \, C_{\mathbf{vv}jj}(t)}}} = \pi C_{\mathbf{vv}jj}(\omega=0).} Computationally, the MSD is calculated by calculating the position autocorrelation since -from Eq. :math:numref:`pfx14` +from Eq. :math:numref:`pfx3` .. math:: - :label: pfx22 + :label: pfx8 - \Delta_{i}^{2}(t) = \left\langle [\mathbf{r}_{i}( t ) - \mathbf{r}_{i}(0)]^2 \right\rangle = \left\langle \mathbf{r}_{i}^{2}(t) \right\rangle + \left\langle \mathbf{r}_{i}^{2}( 0 ) \right\rangle - 2\left\langle \mathbf{r}_{i}(t )\mathbf{r}_{i}(0) \right\rangle + \Delta_{j}^{2}(t) = \left\langle [\mathbf{r}_{j}( t ) - \mathbf{r}_{j}(0)]^2 \right\rangle = \left\langle \mathbf{r}_{j}^{2}(t) \right\rangle + \left\langle \mathbf{r}_{j}^{2}( 0 ) \right\rangle - 2\left\langle \mathbf{r}_{j}(t )\mathbf{r}_{j}(0) \right\rangle -where the last part on the right side Eq. :math:numref:`pfx22` is the position autocorrelation of the particle :math:`i`. +where the last part on the right side Eq. :math:numref:`pfx8` is the position autocorrelation of the particle :math:`j`. .. _analysis-op: @@ -189,90 +196,90 @@ Order Parameter .. _theory-and-implementation-3: - -Adequate and accurate cross comparison of the NMR and MD simulation -data is of crucial importance in versatile studies conformational -dynamics of proteins. NMR relaxation spectroscopy has proven to be a -unique approach for a site-specific investigation of both global -tumbling and internal motions of proteins. The molecular motions -modulate the magnetic interactions between the nuclear spins and lead -for each nuclear spin to a relaxation behaviour which reflects its -environment. Since its first applications to the study of protein -dynamics, a wide variety of experiments has been proposed to investigate -backbone as well as side chain dynamics. Among them, the heteronuclear -relaxation measurement of amide backbone :sup:`15`\ N nuclei is one of -the most widespread techniques. The relationship between microscopic -motions and measured spin relaxation rates is given by Redfield's theory -[Ref13]_. Under the hypothesis that -:sup:`15`\ N relaxation occurs through dipole-dipole interactions with -the directly bonded :sup:`1`\ H atom and chemical shift anisotropy -(CSA), and assuming that the tensor describing the CSA is axially -symmetric with its axis parallel to the N-H bond, the relaxation rates -of the :sup:`15`\ N nuclei are determined by a time correlation -function, - -.. math:: - :label: pfx34 - - {C_{\mathit{ii}}{(t) = \left\langle {P_{2}\left( {\mu_{i}(0)\cdot\mu_{i}(t)} \right)} \right\rangle}} - -which describes the dynamics of a unit vector :math:`\mu_{i}(t)` pointing -along the :sup:`15`\ N-:sup:`1`\ H bond of the residue :math:`i` in the -laboratory frame. Here :math:`P_{2}(x)` is the second order Legendre -polynomial. The Redfield theory shows that relaxation measurements probe -the relaxation dynamics of a selected nuclear spin only at a few -frequencies. Moreover, only a limited number of independent observables -are accessible. Hence, to relate relaxation data to protein dynamics one -has to postulate either a dynamical model for molecular motions or a -functional form for :math:`C_{ii}(t)`, yet depending on a limited number -of adjustable parameters. Usually, the tumbling motion of proteins in -solution is assumed isotropic and uncorrelated with the internal -motions, such that: - -.. math:: - :label: pfx35 - - {C_{\mathit{ii}}{(t) = C^{\mathrm{G}}}(t) C_{\mathit{ii}}^{\mathrm{I}}(t)} - -where :math:`C^{\mathrm{G}}(t)` and :math:`C_{\mathit{ii}}^{\mathrm{I}}(t)` denote the -global and the internal time correlation function, -respectively. Within the so-called model free approach -[Ref14]_, [Ref15]_ -the internal correlation function is modelled by an exponential, - -.. math:: - :label: pfx37 - - {C_{\mathit{ii}}^{\mathrm{I}}{(t) = {S_{i}^{2} + \left( {1 - S_{i}^{2}} \right)}}\exp\left( \frac{- t}{\tau_{\mathrm{eff},i}} \right)} - -Here the asymptotic value - -.. math:: - :label: pfx38 - - {S_{i}^{2} = C_{\mathit{ii}}}\left( {+ \infty} \right) - -\ is the so-called generalized order parameter, which indicates the -degree of spatial restriction of the internal motions of a bond vector, -while the characteristic time :math:`\tau_{\mathrm{eff},i}` is an -effective correlation time, setting the time scale of the -internal relaxation processes. :math:`S_{i}^{2}` can adopt values -ranging from :math:`0` (completely disordered) to :math:`1` (fully ordered). So, -:math:`S_{i}^{2}` is the appropriate indicator of protein backbone motions in -computationally feasible timescales as it describes the spatial aspects -of the reorientational motion of N-H peptidic bonds vector. - -When performing order parameter analysis, MDANSE computes for each -residue :math:`i` both :math:`C_{\mathit{ii}}(t)` and :math:`S_{i}^{2}`. -It also computes a correlation function averaged over all the selected -bonds defined as: - -.. math:: - :label: pfx44 - - {C^{\mathrm{I}}{(t) = {\sum\limits_{i = 1}^{N_{\mathrm{bonds}}}{C_{\mathit{ii}}^{\mathrm{I}}(t)}}}} - -where :math:`N_{\mathrm{bonds}}` is the number of selected bonds for the analysis. +.. note:: + + **This job is under development MDANSE and is currently not available. + The documentation here is out-dated and only left here for referencing + purposes.** + + Adequate and accurate cross comparison of the NMR and MD simulation + data is of crucial importance in versatile studies conformational + dynamics of proteins. NMR relaxation spectroscopy has proven to be a + unique approach for a site-specific investigation of both global + tumbling and internal motions of proteins. The molecular motions + modulate the magnetic interactions between the nuclear spins and lead + for each nuclear spin to a relaxation behaviour which reflects its + environment. Since its first applications to the study of protein + dynamics, a wide variety of experiments has been proposed to investigate + backbone as well as side chain dynamics. Among them, the heteronuclear + relaxation measurement of amide backbone :sup:`15`\ N nuclei is one of + the most widespread techniques. The relationship between microscopic + motions and measured spin relaxation rates is given by Redfield's theory + [Ref13]_. Under the hypothesis that + :sup:`15`\ N relaxation occurs through dipole-dipole interactions with + the directly bonded :sup:`1`\ H atom and chemical shift anisotropy + (CSA), and assuming that the tensor describing the CSA is axially + symmetric with its axis parallel to the N-H bond, the relaxation rates + of the :sup:`15`\ N nuclei are determined by a time correlation + function, + + .. math:: + + {C_{\mathit{ii}}{(t) = \left\langle {P_{2}\left( {\mu_{i}(0)\cdot\mu_{i}(t)} \right)} \right\rangle}} + + which describes the dynamics of a unit vector :math:`\mu_{i}(t)` pointing + along the :sup:`15`\ N-:sup:`1`\ H bond of the residue :math:`i` in the + laboratory frame. Here :math:`P_{2}(x)` is the second order Legendre + polynomial. The Redfield theory shows that relaxation measurements probe + the relaxation dynamics of a selected nuclear spin only at a few + frequencies. Moreover, only a limited number of independent observables + are accessible. Hence, to relate relaxation data to protein dynamics one + has to postulate either a dynamical model for molecular motions or a + functional form for :math:`C_{ii}(t)`, yet depending on a limited number + of adjustable parameters. Usually, the tumbling motion of proteins in + solution is assumed isotropic and uncorrelated with the internal + motions, such that: + + .. math:: + + {C_{\mathit{ii}}{(t) = C^{\mathrm{G}}}(t) C_{\mathit{ii}}^{\mathrm{I}}(t)} + + where :math:`C^{\mathrm{G}}(t)` and :math:`C_{\mathit{ii}}^{\mathrm{I}}(t)` denote the + global and the internal time correlation function, + respectively. Within the so-called model free approach + [Ref14]_, [Ref15]_ + the internal correlation function is modelled by an exponential, + + .. math:: + + {C_{\mathit{ii}}^{\mathrm{I}}{(t) = {S_{i}^{2} + \left( {1 - S_{i}^{2}} \right)}}\exp\left( \frac{- t}{\tau_{\mathrm{eff},i}} \right)} + + Here the asymptotic value + + .. math:: + + {S_{i}^{2} = C_{\mathit{ii}}}\left( {+ \infty} \right) + + \ is the so-called generalized order parameter, which indicates the + degree of spatial restriction of the internal motions of a bond vector, + while the characteristic time :math:`\tau_{\mathrm{eff},i}` is an + effective correlation time, setting the time scale of the + internal relaxation processes. :math:`S_{i}^{2}` can adopt values + ranging from :math:`0` (completely disordered) to :math:`1` (fully ordered). So, + :math:`S_{i}^{2}` is the appropriate indicator of protein backbone motions in + computationally feasible timescales as it describes the spatial aspects + of the reorientational motion of N-H peptidic bonds vector. + + When performing order parameter analysis, MDANSE computes for each + residue :math:`i` both :math:`C_{\mathit{ii}}(t)` and :math:`S_{i}^{2}`. + It also computes a correlation function averaged over all the selected + bonds defined as: + + .. math:: + + {C^{\mathrm{I}}{(t) = {\sum\limits_{i = 1}^{N_{\mathrm{bonds}}}{C_{\mathit{ii}}^{\mathrm{I}}(t)}}}} + + where :math:`N_{\mathrm{bonds}}` is the number of selected bonds for the analysis. .. _analysis-pacf: @@ -281,21 +288,21 @@ Position AutoCorrelation Function ''''''''''''''''''''''''''''''''' The position autocorrelation function (PACF) is similar to the -velocity autocorrelation function described below. In MDANSE the PACF +velocity autocorrelation function in :ref:`analysis-vacf`. In MDANSE the PACF is calculated relative to the atoms average position over the entire trajectory. The PACF of atom type :math:`\alpha` is .. math:: - :label: pfx44a + :label: pfx9 - \mathrm{PACF}_{\alpha}(t) = \frac{1}{3}\frac{W_\alpha}{N_{\alpha}} \sum_{i}^{N_{\alpha}} \left\langle {\Delta \mathbf{r}_{i}(0)\cdot \Delta \mathbf{r}_{i}(t)} \right\rangle + \mathrm{PACF}_{\alpha}(t) = \frac{1}{3}\frac{1}{Nc_{\alpha}} \sum_{j}^{N_{\alpha}} \left\langle {\Delta \mathbf{r}_{j}(0)\cdot \Delta \mathbf{r}_{j}(t)} \right\rangle where .. math:: - :label: pfx44b + :label: pfx10 - \Delta \mathbf{r}_{i}\left( t \right) = \mathbf{r}_{i}(t) - \langle \mathbf{r}_{i}(t_0) \rangle_{t_0} + \Delta \mathbf{r}_{j}\left( t \right) = \mathbf{r}_{j}(t) - \langle \mathbf{r}_{j}(t') \rangle_{t'} so that the origin dependence of the PACF function is removed. @@ -304,37 +311,57 @@ so that the origin dependence of the PACF function is removed. Van Hove Function ''''''''''''''''' The van Hove function describes the probability of finding a particle -:math:`j` at time :math:`t` with a displacement of :math:`\mathbf{r}` from a -particle :math:`i` at a time :math:`0` +:math:`k` at time :math:`t` with a displacement of :math:`\mathbf{r}` from a +particle :math:`j` at a time :math:`0` .. math:: - G(\mathbf{r}, t) = \frac{1}{N} \sum_{i}^{N}\sum_{j}^{N} \left\langle \delta [\mathbf{r} - \mathbf{r}_{j}(t) + \mathbf{r}_{i}(0)] \right\rangle. + :label: pfx11 + + G(\mathbf{r}, t) = \frac{1}{N} \sum_{jk} \left\langle \delta [\mathbf{r} - \mathbf{r}_{k}(t) + \mathbf{r}_{j}(0)] \right\rangle. The van Hove function is related to the intermediate scattering function via a Fourier transform and the dynamic structure factor via a double Fourier transform .. math:: - F(\mathbf{q}, t) &= \int \mathrm{d}\mathbf{r} \, G(\mathbf{r},t) \exp(i \mathbf{q} \cdot \mathbf{r}) \\ - S(\mathbf{q}, \omega) &= \int \mathrm{d}t \int \mathrm{d}\mathbf{r} \, G(\mathbf{r},t) \exp(i \mathbf{q} \cdot \mathbf{r} - i \omega t) + :label: pfx12 + + F(\mathbf{q}, t) = \int \mathrm{d}\mathbf{r} \, G(\mathbf{r},t) e^{i \mathbf{q} \cdot \mathbf{r}} + +.. math:: + :label: pfx13 + + S(\mathbf{q}, \omega) = \int \mathrm{d}t \int \mathrm{d}\mathbf{r} \, G(\mathbf{r},t) e^{i \mathbf{q} \cdot \mathbf{r} - i \omega t} and can be split into distinct and self parts where .. math:: - G_{\mathrm{d}}(\mathbf{r}, t) &= \frac{1}{N} \sum_{i}^{N}\sum_{j \neq i}^{N} \left\langle \delta [\mathbf{r} - \mathbf{r}_{j}(t) + \mathbf{r}_{i}(0)] \right\rangle \\ - G_{\mathrm{s}}(\mathbf{r}, t) &= \frac{1}{N} \sum_{i} \left\langle \delta [\mathbf{r} - \mathbf{r}_{i}(t) + \mathbf{r}_{i}(0)] \right\rangle + :label: pfx14 + + G_{\mathrm{d}}(\mathbf{r}, t) = \frac{1}{N} \sum_{j}\sum_{k \neq j} \left\langle \delta [\mathbf{r} - \mathbf{r}_{k}(t) + \mathbf{r}_{j}(0)] \right\rangle + +.. math:: + :label: pfx15 + + G_{\mathrm{s}}(\mathbf{r}, t) = \frac{1}{N} \sum_{j} \left\langle \delta [\mathbf{r} - \mathbf{r}_{j}(t) + \mathbf{r}_{j}(0)] \right\rangle In MDANSE the distinct and self parts of the van Hove function are -spherically averaged and normalised so that -:math:`\mathcal{G}_{\mathrm{d}}(r, t) = G_{\mathrm{d}}(r, t) / \rho` and -:math:`\mathcal{G}_{\mathrm{s}}(r, t) = G_{\mathrm{s}}(r, t) / \rho`. -For liquid or gaseous systems +spherically averaged and normalised so that for liquid or gaseous systems + +.. math:: + :label: pfx16 + + \lim_{r \rightarrow \infty } \mathcal{G}_{\mathrm{d}}(r, t) = \lim_{t \rightarrow \infty } \mathcal{G}_{\mathrm{d}}(r, t) = 1 \\ .. math:: - &\lim_{r \rightarrow \infty } \mathcal{G}_{\mathrm{d}}(r, t) = \lim_{t \rightarrow \infty } \mathcal{G}_{\mathrm{d}}(r, t) = 1 \\ - &\lim_{t \rightarrow \infty } \mathcal{G}_{\mathrm{s}}(r, t) = N^{-1} + :label: pfx17 + + \lim_{t \rightarrow \infty } \mathcal{G}_{\mathrm{s}}(r, t) = N^{-1} -where in the thermodynamic limit :math:`N \rightarrow \infty`. +where +:math:`\mathcal{G}_{\mathrm{d}}(r, t) = G_{\mathrm{d}}(r, t) / \rho` and +:math:`\mathcal{G}_{\mathrm{s}}(r, t) = G_{\mathrm{s}}(r, t) / \rho`; +in the thermodynamic limit :math:`N \rightarrow \infty`. .. _analysis-vacf: @@ -368,53 +395,42 @@ oscillation before decaying to zero. This decaying time can be considered as the average time for a collision between two atoms to occur before they diffuse away. -Mathematically, the VACF of atom :math:`\alpha` in an atomic or molecular system is +Mathematically, the VACF of atom :math:`j` in an atomic or molecular system is usually defined as .. math:: - :label: pfx45 + :label: pfx18 - {C_{\mathit{vv};\mathit{\alpha\alpha}}(t)\doteq\frac{1}{3}\left\langle {v_{\alpha}\left( 0 \right)\cdot v_{\alpha}\left( t \right)} \right\rangle.} + {C_{\mathbf{vv}jj}(t) = \frac{1}{3}\left\langle {\mathbf{v}_{j}( 0 )\cdot \mathbf{v}_{j}( t )} \right\rangle.} In some cases, e.g. for non-isotropic systems, it is useful to define VACF along a given axis, .. math:: - :label: pfx46 + :label: pfx19 - {C_{\mathit{vv};\mathit{\alpha\alpha}}\left( {t;n} \right)\doteq\left\langle {v_{\alpha}\left( {0;n} \right)v_{\alpha}\left( {t;n} \right)} \right\rangle,} + {C_{\mathbf{vv}jj}(t, \hat{\mathbf{n}}) = \frac{1}{3}\left\langle {v_{j}( 0, \hat{\mathbf{n}}) v_{j}( t, \hat{\mathbf{n}})} \right\rangle \qquad v_{j}(t, \hat{\mathbf{n}}) = \hat{\mathbf{n}} \cdot \mathbf{v}_{j}(t)} -where :math:`v_{\alpha}(t; n)` is given by +where the vector :math:`\hat{\mathbf{n}}` is a unit vector defining a space-fixed +axis. The VACF of the particles in a many-body system can be related to the +incoherent dynamic structure factor by the relation .. math:: - :label: pfx47 - - {v_{\alpha}\left( {t;n} \right)\doteq n\cdot v_{\alpha}(t).} - -The vector :math:`n` is a unit vector defining a space-fixed axis. - -The VACF of the particles in a many body system can be related to the -incoherent dynamic structure factor by the relation: + :label: pfx20 -.. math:: - :label: pfx48 + {\lim\limits_{q\rightarrow 0}\frac{1}{3}\frac{\omega^{2}}{q^{2}}S_{\mathrm{inc}}{\left( {q,\omega} \right) = \mathrm{DOS}}(\omega, \hat{\mathbf{q}}).} - {\lim\limits_{q\rightarrow 0}\frac{\omega^{2}}{q^{2}}S{\left( {q,\omega} \right) = G}(\omega),} -where :math:`G(\omega)` is the density of states. For an isotropic system it -reads +where :math:`\hat{\mathbf{q}}` is the unit vector in the direction of :math:`\mathbf{q}`. +Here the density of states is as weight sum of the Fourier transform of +the projected VACF .. math:: - :label: pfx49 + :label: pfx21 - {G{(\omega) = {\sum\limits_{\alpha}{b_{\alpha,\mathit{inc}}^{2}{\overset{\sim}{C}}_{\mathit{vv};\mathit{\alpha\alpha}}(\omega)}}},} + {\mathrm{DOS}{(\omega, \hat{\mathbf{q}}) = {\frac{1}{N}\sum\limits_{j}{b_{\mathrm{inc},j}^{2}{{C}}_{\mathbf{vv}jj}(\omega, \hat{\mathbf{q}})}}},} .. math:: - :label: pfx50 - - {{\overset{\sim}{C}}_{\mathit{vv};\mathit{\alpha\alpha}}{(\omega) = \frac{1}{2\pi}}{\int\limits_{- \infty}^{+ \infty}\mathrm{d} t \,}\exp\left\lbrack {{- i}\omega t} \right\rbrack C_{\mathit{vv};\mathit{\alpha\alpha}}(t).} + :label: pfx22 -For non-isotropic systems, relation :math:numref:`pfx48` holds if the DOS -is computed from the atomic velocity autocorrelation -functions :math:`C_{\mathit{vv};\mathit{\alpha\alpha}}\left( {t;n_{q}} \right)` -where :math:`n_q` is the unit vector in the direction of :math:`q`. + {C_{\mathbf{vv}jj}{(\omega, \hat{\mathbf{q}}) = \frac{1}{2\pi}}{\int\limits_{- \infty}^{+ \infty}\mathrm{d} t \,} C_{\mathbf{vv}jj}(t, \hat{\mathbf{q}}) e^{-i \omega t}}. diff --git a/Doc/pages/weights.rst b/Doc/pages/weights.rst index 9dcf617c5..51d337739 100644 --- a/Doc/pages/weights.rst +++ b/Doc/pages/weights.rst @@ -9,12 +9,12 @@ scattering functions are .. math:: :label: ws1 - F_{\text{coh},\alpha\beta}{(\mathbf{q},t) = \frac{W_{\alpha\beta}}{N \sqrt{c_{\alpha}c_{\beta}}}}{\sum\limits_{j}^{N_{\alpha}}{\sum\limits_{k}^{N_{\beta}}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{k}\left( t \right)} \right\rbrack} \right\rangle}}, + \mathcal{F}_{\text{coh},\alpha\beta}{(\mathbf{q},t) = \frac{W_{\alpha\beta}}{N \sqrt{c_{\alpha}c_{\beta}}}}{\sum\limits_{j}^{N_{\alpha}}{\sum\limits_{k}^{N_{\beta}}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{k}\left( t \right)} \right\rbrack} \right\rangle}}, .. math:: :label: ws2 - F_{\text{inc},\alpha}{(\mathbf{q},t ) = \frac{W_{\alpha}}{Nc_{\alpha}}}{\sum\limits_{j}^{N_{\alpha}}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{j}\left( t \right)} \right\rbrack} \right\rangle} + \mathcal{F}_{\text{inc},\alpha}{(\mathbf{q},t ) = \frac{W_{\alpha}}{Nc_{\alpha}}}{\sum\limits_{j}^{N_{\alpha}}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{j}\left( t \right)} \right\rbrack} \right\rangle} where :math:`\alpha` and :math:`\beta` are the atom-types. :math:`W_{\alpha\beta}` and :math:`W_{\alpha}` are the weights of the @@ -27,17 +27,17 @@ and the total number of atoms. The total is now a sum of the partial terms .. math:: :label: ws3 - F_{\text{coh}}(\mathbf{q},t) = \sum_{\alpha}\sum_{\beta \geq \alpha} F_{\text{coh},\alpha\beta}(\mathbf{q},t), + F_{\text{coh}}(\mathbf{q},t) = \sum_{\alpha}\sum_{\beta \geq \alpha} \mathcal{F}_{\text{coh},\alpha\beta}(\mathbf{q},t), .. math:: :label: ws4 - F_{\text{inc}}(\mathbf{q},t) = \sum_{\alpha} F_{\text{inc},\alpha}(\mathbf{q},t). + F_{\text{inc}}(\mathbf{q},t) = \sum_{\alpha} \mathcal{F}_{\text{inc},\alpha}(\mathbf{q},t). Note that for summation involving two atom-types only the unique pairs are summed up. This is because in MDANSE the off-diagonal weight terms are doubled and and we assumed that -:math:`F_{\text{coh},\alpha\beta} = F_{\text{coh},\beta\alpha}`. +:math:`\mathcal{F}_{\text{coh},\alpha\beta} = \mathcal{F}_{\text{coh},\beta\alpha}`. .. _water-dos-weighted: @@ -54,28 +54,28 @@ The partial properties can also be scaled without the weights .. math:: :label: ws5 - \mathcal{F}_{\text{coh},\alpha\beta}{(\mathbf{q},t) = \frac{1}{N c_{\alpha} c_{\beta}}}{\sum\limits_{j}^{N_{\alpha}}{\sum\limits_{k}^{N_{\beta}}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{k}\left( t \right)} \right\rbrack} \right\rangle}}, + F_{\text{coh},\alpha\beta}{(\mathbf{q},t) = \frac{1}{N c_{\alpha} c_{\beta}}}{\sum\limits_{j}^{N_{\alpha}}{\sum\limits_{k}^{N_{\beta}}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{k}\left( t \right)} \right\rbrack} \right\rangle}}, .. math:: :label: ws6 - \mathcal{F}_{\text{inc},\alpha}{(\mathbf{q},t ) = \frac{1}{N c_{\alpha}}}{\sum\limits_{j}^{N_{\alpha}}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{j}\left( t \right)} \right\rbrack} \right\rangle} + F_{\text{inc},\alpha}{(\mathbf{q},t ) = \frac{1}{N c_{\alpha}}}{\sum\limits_{j}^{N_{\alpha}}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{j}\left( t \right)} \right\rbrack} \right\rangle} so the total will now be a weighted sum of these partial terms .. math:: :label: ws7 - F_{\text{coh}}(\mathbf{q},t) = \sum_{\alpha}\sum_{\beta \geq \alpha} W_{\alpha\beta} \mathcal{F}_{\text{coh},\alpha\beta}(\mathbf{q},t), + F_{\text{coh}}(\mathbf{q},t) = \sum_{\alpha}\sum_{\beta \geq \alpha} W_{\alpha\beta} F_{\text{coh},\alpha\beta}(\mathbf{q},t), .. math:: :label: ws8 - F_{\text{inc}}(\mathbf{q},t) = \sum_{\alpha} W_{\alpha} \mathcal{F}_{\text{inc},\alpha}(\mathbf{q},t). + F_{\text{inc}}(\mathbf{q},t) = \sum_{\alpha} W_{\alpha} F_{\text{inc},\alpha}(\mathbf{q},t). -In the MDANSE_GUI you have the option to plot either weighted (e.g. :math:`F_{\text{coh},\alpha\beta}` -and :math:`F_{\text{inc},\alpha}`) or unweighted (e.g. :math:`\mathcal{F}_{\text{coh},\alpha\beta}` -and :math:`\mathcal{F}_{\text{inc},\alpha}`) partial properties. +In the MDANSE_GUI you have the option to plot either weighted (e.g. :math:`\mathcal{F}_{\text{coh},\alpha\beta}` +and :math:`\mathcal{F}_{\text{inc},\alpha}`) or unweighted (e.g. :math:`F_{\text{coh},\alpha\beta}` +and :math:`F_{\text{inc},\alpha}`) partial properties. .. _water-pdf-unweighted: @@ -158,21 +158,21 @@ normalized slightly differently. In MDANSE the (weighted) partial static structu factor (SSF) is .. math:: - :label: ws15 + :label: ws14 - S_{\alpha\beta}(q) = W_{\alpha\beta} \left[ 1 + \frac{4 \pi \rho}{q} \int_{0}^{\infty} \mathrm{d}r \, \left[ g_{\alpha\beta}(r) - 1\right] r\sin(qr)\right] + S_{\alpha\beta}(q) = W_{\alpha\beta} \left[ 1 + \frac{4 \pi \rho}{q} \int\limits_{0}^{\infty} \mathrm{d}r \, \left[ g_{\alpha\beta}(r) - 1\right] r\sin(qr)\right] where .. math:: - :label: ws16 + :label: ws15 g_{\alpha\beta}(r) = \frac{1}{N c_{\alpha} c_{\beta}} \sum_{j}^{N_\alpha} \sum_{k\neq j}^{N_\beta} \left\langle \delta(r - \vert \mathbf{r}_{k} + \mathbf{r}_{j} \vert ) \right\rangle are the partial PDFs. Using ``b_coherent``, the weights are .. math:: - :label: ws17 + :label: ws16 W_{\alpha\beta} = \left[2 - \delta_{\alpha\beta}\right]\frac{c_{\alpha}c_{\beta} b_{\mathrm{coh},\alpha}b_{\mathrm{coh},\beta}}{\sum_{\gamma\delta} c_{\gamma}c_{\delta} b_{\mathrm{coh},\gamma}b_{\mathrm{coh},\delta}}.