Fluctuation-dissipation theorem, kinetic stochastic integral and
efficient simulations
Markus HuŽ tter and Hans Christian O ttinger
ET H-ZuŽ rich, Department of Materials, Institute of Polymers, CH-8092 ZuŽ rich, Switzerland
Di†usive systems respecting the Ñuctuation-dissipation theorem with multiplicative noise have been studied on the level of
stochastic di†erential equations. We propose an efficient simulation scheme motivated by the direct deÐnition of the “ kinetic
stochastic integral Ï, which di†ers from the better known Itoü and the Stratonovich integrals. This simulation scheme is based on
introducing the identity matrix, expressed in terms of the di†usion tensor and its inverse, in front of the noise term, and evaluating these factors at di†erent times.
related to the di†usion tensor by
1 Introduction
Di†usion equations are studied whenever Ñuctuations, or irreversible behaviour in general, are involved ; examples being
particle di†usion, heat conduction, viscous Ñow and the
description of complex Ñuids. For a d-dimensional variable x,
these equations can be written in the form
C
D
d
d
d
1 d
p(x, t)
p(x, t) \ [ É [A(x, t)p(x, t)] ]
É D (x, t) É
dx
dt
dx
2 dx
(1)
where the term containing the d-dimensional vector A is
called drift, and the other containing the positive semi-deÐnite
d ? d-matrix D is called the di†usion term. For the following,
it is essential to notice that the di†usion tensor D is placed
between the two partial derivatives. In systems where the drift
originates from a gradient of a potential that is mediated
through the di†usion or mobility tensor, i.e. A is of the form
C
D
d
1
/(x, t)
A(x, t) \ [ D (x, t) É
dx
2
(2)
one then Ðnds that the “ Boltzmann distribution Ï
p(x) B exp[[/(x)] is a stationary solution of the above di†usion equation [eqn. (1)].
A solution of the di†usion equation can often not be found
in closed form if the di†usion tensor depends on the state of
the system, i.e. for multiplicative noise, and one is led to solve
the di†usion equation numerically with Ðnite element
methods. However, for many complex systems, such as
polymer solutions or colloidal suspensions, the large number
of degrees of freedom is prohibitive for a numerical solution of
the di†usion equation. Rather than studying the system by
means of the distribution function p(x, t) for the variable x,
one thus looks for a set of trajectories x which have the distribution p(x, t), meaning that one tries to solve stochastic differential equations.1,2 A trajectory, which is a sample for the
di†usion equation [eqn. (1)], is given by the following stochastic di†erential equation1h3 (Itoü version, denoted by the symbol
“ I Ï) :
C
dx \ A(x, t) ]
A
BD
1 d
É D (x, t)
2 dx
dt ] B (x, t) I dW
(3)
where the random increments are composed of d@-dimensional
Wiener increments dW and a d ? d@-matrix B , the latter being
B (x, t) É B (x, t)T \ D (x, t)
(4)
where superscript T denotes the “ transpose Ï of matrix B . Eqn.
(4) relating random forces and di†usive/dissipative dynamics is
the well known Ñuctuation-dissipation theorem of the second
kind,1,4 which is a key result of statistical mechanics. Choosing the Itoü version of the stochastic di†erential eqn. (3) means
that the random increments B (x, t) I dW have mean value
zero. Notice the divergence term in eqn. (3) : putting the di†usion tensor in eqn. (1) between the two derivatives is the
reason for the divergence term in eqn. (3). In other words, for
systems given by eqn. (1) with a drift term given by eqn. (2),
the Ñuctuation-dissipation theorem and the Boltzmann stationary solution lead to an ItoüÈstochastic di†erential equation
with this additional contribution to the drift term. We here
propose a numerical integration scheme for stochastic di†erential equations given by eqn. (3), that circumvents the calculation of the divergence term completely, but rather constructs
it by using a two-step scheme with the main goal of substantially reducing the computational needs in simulations. In
order to motivate this integration scheme we Ðrst deÐne a new
stochastic integral, from which we then heuristically show
how the divergence term is constructed.
One should mention that a stochastic integral deÐnition,
which naturally respects the Ñuctuation-dissipation theorem,
has been given already for the one-dimensional case by
Tsekov,5 but we give here a non-trivial generalization to arbitrary dimensions. The latter stochastic integral shall be called
“ kinetic Ï, as proposed by Klimontovich.6,7
2 Multi-dimensional deÐnition of the “ kinetic
stochastic integral Ï
There are currently three di†erent deÐnitions for stochastic
integrals, the most famous of which are the Itoü and Stratonovich versions,2,3 respectively. While the transformations
among di†erent types of integrals are straightforward in principle, certain integrals may be much more convenient for
developing efficient simulation algorithms. They di†er by the
time point at which the integrand is evaluated when writing
the integral as a sum over Ðnite time intervals : Itoü evaluates
the integrand at the starting point of the time interval,
whereas Stratonovich uses the midpoint values. A third stochastic integral deÐnition has recently been given by Tsekov,5
using the endpoint values of the integrand. He proved this
integral to respect the Ñuctuation-dissipation theorem naturally in the one-dimensional case. However, it does not
J. Chem. Soc., Faraday T rans., 1998, 94(10), 1403È1405
1403
possess this desirable property in the multi-dimensional case.
We give here a deÐnition for the kinetic stochastic integral
with the main property of naturally respecting the Ñuctuationdissipation theorem, i.e. of incorporating the divergence term
in eqn. (3), also for an arbitrary multi-dimensional case. A
deÐnition of this new integral, which shall be denoted by the
symbol ), in terms of the Itoü integral by
P
P
B (x) ) dW À B (x) I dW ]
1
2
PC
D
d
É D (x) dt
dx
does not provide us with any more insight for the construction of the desired integration scheme. Much more useful is
the key result of this paper, which is the deÐnition of this integral in terms of the mean-square limit of a time-discretized
sum,
P
M
B (x) ) dW À ms [ lim ; 1 [D (x ) É D (x ) ] 1]
2
ti`1
inv ti
M?= i/1
(5)
É B (x ) É [W [ W ]
ti
ti`1
ti
with D À B É B T and the Wiener increments *W À W
ti
ti`1
[ W having mean value S*W T \ 0 and variance
ti
ti
S*W *W T \ d *t1. The deÐnition of the inverse D
has
tk
ik
inv
ti
to account for the fact that the null-space of D might be nontrivial, i.e. that some linear combinations of the variables x are
not a†ected by the random and di†usive motion. We therefore
deÐne the matrix D to be the null-matrix on the null-space
inv
of D and the inverse of D on the remaining subspace, i.e. the
inverse of the diagonalized matrix D is deÐned by taking the
inverse of the eigenvalues with the deÐnition 1 À 0. (As a con0
sequence, the null-spaces of D and D are identical at a given
inv
time t ). It is essential that D is evaluated at the same time as
inv
i
B for the image of B and the image of D to be the same subspace. Furthermore, this makes clear that the deÐnition 1 À 0
0
on the null-space does not lead to any complications and is
not related to any physical singularities, but only stands for
excluding the non-di†usive subspace from the deÐnition of the
stochastic integral. For a strictly positive-deÐnite di†usion
tensor, D is simply the inverse of D .
inv
One can show heuristically that the above deÐnition [eqn.
(5)] for the kinetic stochastic integral is indeed the sum of the
Itoü integral and the desired divergence term by making a
Taylor expansion of the Ðrst di†usion tensor in eqn. (5) and
expressing the end-point value x
in terms of the startpoint
ti`1
value x through the stochastic integral eqn. (3) to Ðrst order
ti
in *W . One Ðnds in the mean-square sense
ti
d
1
É D (x) dt
B (x) ) dW \ B (x) I dW ]
(6)
dx
2
P
P
PC
D
illustrating the existence of the non-random contribution in
the kinetic integral, as the expectation value SB (x) ) dWT \
S(d/dx) É D (x)Tdt is non-vanishing in general. We point out
that the factorization in the integrand and the evaluation of
the integrand factors at di†erent times are essential for
producing the divergence term. The occurrence of the divergence term when going to non-vanishing time steps, i.e. Ðnite
time resolution, was used by Fixman8 when constructing
Langevin equations and the numerical integration scheme for
the simulation of polymer dynamics, but it was not formulated in terms of stochastic integration.
3 Construction of a numerical integration scheme
It has been shown in the previous section that the di†erences
between the Itoü and the kinetic versions of the stochastic integral is the inclusion of the extra divergence term in the kinetic
integral, formally changing the equations for dx but leaving
the solution x unchanged, since
1404
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
G
A(x) ]
A
BH
1 d
Æ D (x)
2 dx
dt ] B (x) I dW
\ A(x) dt ] B (x) ) dW
The main advantage of this kinetic integral deÐnition lies,
however, in the possibility of constructing efficient numerical
integration schemes for stochastic processes as in eqn. (3),
without ever calculating the divergence of the di†usion tensor.
Such integration schemes have already been used in Brownian
dynamics simulations of dense colloidal suspensions,9 being of
Ðrst order in the time step *t. However, to the best of our
knowledge, no direct mathematical deÐnition of a kinetic stochastic integral for arbitrary dimensions has ever been given.
We think that a deÐnition by means of stochastic integrals,
such as eqn. (5), is necessary to derive more efficient numerical
integration schemes, especially to develop higher order integration schemes.
In the following, we heuristically derive a weak2 Ðrst order
numerical integration scheme from our kinetic integral deÐnition. Writing down the kinetic version of the stochastic di†erential equation,
dx \ A(x ) dt ] B (x ) ) dW
t
t
t
t
= A(x ) dt ] 1 [D (x ] dx ) É D (x ) ] 1] É B (x ) É dW (7)
inv t
t
2
t
t
t
t
suggests the following two-step scheme for the numerical integration. The predictor step is given by
(8)
*xp \ A(x )*t ] B (x ) É *W
ti
ti
ti
ti
where *W \ W [ W are the Wiener increments with
ti
ti`1
i
mean value S*W T \ 0 and variance S*W *W T \ d *t1.
ti
tk
ik
ti
This predictor value is now used in the Ðrst di†usion tensor
term on the second line of eqn. (7) for the corrector step :
*xc \ 1 [A(x ] *xp) ] A(x )]*t
ti
ti
ti
ti 2
] 1 [D (x ] *xp) É D (x ) ] 1] É B (x ) É *W
(9)
2
ti
ti
ti
ti
inv ti
It can be shown that the above two-step integration scheme is
weakly convergent to Ðrst order in the time step *t by
expanding D (x ] *xp) up to Ðrst order in *W . For a conti
ti
ti
stant di†usion tensor, the above scheme is even weakly convergent to second order. The term involving D (x ] *xp)
ti
ti
replaces the calculation of the divergence of the di†usion
tensor in the Itoü stochastic di†erenctial eqn. (3). Thus, the
kinetic version is to be favoured to the Itoü version, except in
cases where a closed expression for (d/dx) É D (x) can be given.
For cases where only D
but not D itself is explicitly
inv
known (the connection between D and D being as described
inv
in Section 2), the determination of B and especially of
(d/dx) Æ D (x) is computationally very time consuming.
However, the deÐnition of the kinetic stochastic integral can
be reformulated in a way to circumvent this problem : using
D É D É B \ B , we Ðnd with B À D É B from the deÐnition
inv
inv
inv
of the kinetic stochastic integral
P
M
B (x) ) dW \ ms [ lim ; 1 [D (x ) ] D (x )]
2
ti`1
ti
M?= i/1
(10)
É B (x ) É [W [ W ]
tia1
ti
inv ti
with B É B T \ D É B É B T É D T \ D É D É D T \D T \ D .
inv inv
inv
inv
inv
inv
inv
inv
For such systems, the predictor step is then put into the form
*xp \ A(x )*t ] D (x ) É B (x ) É *W
ti
ti
ti
inv ti
ti
and the corrector step is given by
*xc \ 1 [A(x ] *xp) ] A(x )]*t
ti
ti
ti
ti 2
] 1 [D (x ] *xp) ] D (x )] É B (x ) É *W
ti
ti
ti
ti
inv ti
2
(11)
(12)
The contributions to *xp and *xc in eqn. (11) and (12), respecti
ti
tively, which involve the di†usion tensor D , can then be solved
for by using the tensor D and a conjugate gradient method.
inv
In this sense, it is crucial to note that the integral in eqn. (10)
is only well deÐned if the null-space of D at t
is equal to,
inv
i`1
or a subset of, the null-space at t ; this was not a restriction to
i
eqn. (5). Again, the term involving D (x ] *xp) replaces the
ti
ti
calculation of the divergence of the di†usion tensor. This latter
integration scheme has been used by Ball and Melrose in
Brownian dynamics simulations of dense colloidal suspensions,9 where the time invariance of the null-space of D is
inv
motivated by the Galilean invariance of the equations of
motions (i.e. total particle momentum lies in the null-space of
D for all times).
inv
It should be recalled that the integration scheme proposed
in eqn. (8) and (9) applies to a larger class of problems than
that in eqn. (11) and (12), as the former requires no restrictions
on the null-space of D
or D . The deÐnition of the kinetic
inv
integral in eqn. (5) [and not eqn. (10)] is thus to be considered
as the starting point when developing numerical integration
schemes.
4 Conclusions
A deÐnition for a stochastic integral was given that naturally
respects the Ñuctuation dissipation theorem by including the
divergence of the di†usion tensor in the arbitrarily multidimensional case. It has been shown that this integral deÐnition can be used to construct numerical integration schemes
for stochastic di†erential equations without ever calculating
the divergence of the di†usion tensor and thereby substantially reducing the computational needs.
This work was supported by grant no. 2100-046728.96/1 from
the Swiss National Science Foundation.
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1405