51 lines
2.5 KiB
ReStructuredText
51 lines
2.5 KiB
ReStructuredText
Evaluation of dynamic properties
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================================
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Dynamic properties like mean square displacement are calculated with the
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function :func:`mdevaluate.correlation.shifted_correlation`.
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This function takes a correlation function and calculates the averaged
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time series of it, by shifting a time interval over the trajectory.
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::
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from mdevaluate import correlation
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time, msd_amim = correlation.shifted_correlation(correlation.msd, com_amim, average=True)
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plot(time,msd_amim)
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The result of :func:`shifted_correlation` are two lists, the first one (``time``)
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contains the times of the frames that have been used for the correlation.
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The second list ``msd_amim`` is the correlation function at these times.
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If the keyword ``average=False`` is given, the correlation function for each shifted
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time window will be returned.
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Arguments of ``shifted_correlation``
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------------------------------------
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The function :func:`mdevaluate.correlation.shifted_correlation` accepts several keyword arguments.
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With those arguments, the calculation of the correlation function may be controlled in detail.
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The mathematical expression for a correlation function is the following:
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.. math:: S(t) = \frac{1}{N} \sum_{i=1}^N C(f, R, t_i, t)
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Here :math:`S(t)` denotes the correlation function at time t, :math:`R` are the coordinates of all atoms
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and :math:`t_i` are the onset times (:math:`N` is the number of onset times or time windows).
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Note that the outer sum and division by :math:`N` is only carried out if ``average=True``.
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The onset times are defined by the keywords ``segments`` and ``window``, with
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:math:`N = segments` and :math:`t_i = \frac{ (1 - window) \cdot t_{max}}{N} (i - 1)` with the total simulation time :math:`t_{max}`.
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As can be seen ``segments`` gives the number of onset times and ``window`` defines the part of the simulation time the correlation is calculated for,
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hence ``window - 1`` is the part of the simulation the onset times a distributed over.
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:math:`C(f, R, t_0, t)` is the function that actually correlates the function :math:`f`.
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For standard correlations the functions :math:`C(...)` and :math:`f` are defined as:
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.. math:: C(f, R, t_0, t) = f(R(t_0), R(t_0 + t))
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.. math:: f(r_0, r) = \langle s(r_0, r) \rangle
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Here the brackets denote an ensemble average, small :math:`r` are coordinates of one frame and :math:`s(r_0, r)` is the value that is correlated,
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e.g. for the MSD :math:`s(r_0, r) = (r - r_0)^2`.
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The function :math:`C(f, R, t_0, t)` is specified by the keyword ``correlation``, the function :math:`f(r_0, r)` is given by ``function``.
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