Chemical Physics 370 (2010) 20–28

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Chemical Physics

journal homepage: www.elsevier .com/locate /chemphys

Solving evolution equations using interacting trajectory ensembles

Patrick Hogan a, Adam Van Wart a, Arnaldo Donoso b, Craig C. Martens a,*

a Department of Chemistry, University of California, Irvine, CA 92697-2025, USAb Laboratorio de Física Estadística de Sistemas Desordenados, Centro de Física, Instituto Venezolano de Investigaciones Científicas, IVIC, Caracas, Venezuela

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 September 2009In final form 16 December 2009Available online 24 December 2009

Keywords:Molecular dynamicsQuantum dynamicsTrajectory methodsStochastic methods

0301-0104/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.chemphys.2009.12.023

* Corresponding author. Tel.: +1 949 824 8768; faxE-mail address: cmartens@uci.edu (C.C. Martens).

In this paper, we describe a general approach to solving evolution equations for probability densitiesusing interacting trajectory ensembles. Assuming the existence of a positive definite (probabilistic)description of the state of the system, we derive general equations of motion for the trajectories in thekinematic space (e.g., configuration or phase space). The vector field describing the time rate of changeof the trajectory ensemble members depends, in general, on both external forces and on the probabilitydensity itself. The dependence of the equations of motion on the probability density lead to interactionsbetween the ensemble members and a loss of their statistical independence. The formalism is illustratedby a number of numerical examples. For multidimensional systems, a gauge-like freedom exists in thechoice of the underlying vector field, which leaves the evolution of the probability density invariant.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

The trajectory is a central object in the theoretical description ofthe time evolution of particles. In classical mechanics [1], the posi-tion x of a particle of mass m evolves with time under the forcesdefining the dynamical system. The function xðtÞ then traces outthe trajectory of the particle in the configuration space in the New-tonian or Lagrangian formulations of dynamics, while the timederivative vðtÞ ¼ _xðtÞ defines the instantaneous velocity. In Hamil-tonian dynamics, the corresponding trajectory xðtÞ ¼ ðqðtÞ; pðtÞÞmoves in the system’s phase space, with the functions qðtÞ andpðtÞ solving Hamilton’s equations [1],

_q ¼ @H@p

; ð1Þ

_p ¼ � @H@q

; ð2Þ

where Hðq;pÞ is the Hamiltonian of the system.The classical mechanics of deterministic systems can be gener-

alized to incorporate the effect of many external ‘‘bath” degrees offreedom in the form of friction and random forces; an example isthe Langevin equation [2–4]

m€x ¼ �U0ðxÞ �mco _xþ f ðtÞ; ð3Þ

where U0ðxÞ is the derivative of the potential energy, co is the fric-tion constant, and f ðtÞ is a random force. The force and frictionare related by the second fluctuation-dissipation theorem:

ll rights reserved.

: +1 949 824 8571.

hf ð0Þf ðtÞi ¼ 2kBTcodðtÞ; ð4Þ

where kB is Boltzmann’s constant, and the average on the left is overrealizations of the random force. In this case, the correlation func-tion of the force is a d-function, indicating Markovian (memoryless)stochastic dynamics; this can be generalized to include finite mem-ory effects. The trajectory xðtÞ that solves Eq. (3) is itself is a randomfunction of time. In Ref. [5], Pollak describes the connection be-tween infinite dimensional Hamiltonian systems and reduced sto-chastic descriptions, such as the generalized Langevin equation(see also [6]).

In classical mechanics, an alternative view of the dynamics of asystem is given by considering statistical distributions in the sys-tem’s configuration or phase space, consistent with constraintson the system. These distributions connect with the trajectorydescription as representations of ensembles of trajectories evolvingin time with their initial conditions sampled from the initial distri-bution. This viewpoint is important for describing the equilibriumor non-equilibrium statistical mechanics of many-body systems,where the number of degrees of freedom is much higher thanthe measurable and controllable macroscopic parameters, such astemperature and volume [2–4]. A single trajectory in such casescontains too much microscopic detail and is statistically insignifi-cant, so the distribution itself is the central quantity.

For classical deterministic systems, the dynamics are deter-mined by the classical Liouville equation [2]

@q@t¼ fH;qg; ð5Þ

where qðq;p; tÞ is the probability density in phase space, Hðq; pÞ isthe system Hamiltonian, and fH;qg is the Poisson bracket

P. Hogan et al. / Chemical Physics 370 (2010) 20–28 21

fH;qg ¼ @H@q

@q@p� @q@q

@H@p

: ð6Þ

For a classical system in contact with a thermal environment,the generalization analogous to the Langevin equation (3) is theKramers equation [2–4],

@q@t¼ fH;qg þ co

@

@ppþmkBT

@

@p

� �q: ð7Þ

The stochastic force appearing in the trajectory representation, Eq.(3), is here replaced by a diffusive but deterministic term.

The state of a quantum mechanical system is described by itswave function wðx; tÞ, whose absolute square jwðx; tÞj2 gives theconfiguration space probability density, or by the density operatorq̂ðtÞ, whose diagonal element hxjq̂jxi gives the same quantity, evenfor statistical mixed states [7]. The density operator obeys thequantum Liouville equation,

i�hdq̂dt¼ ½bH; q̂�; ð8Þ

where bH is the Hamiltonian operator, ½bH; q̂� is the commutator ofthe two operators, and �h is Planck’s constant divided by 2p.

In phase space, quantum systems can be described by theWigner function [8–11],

qW ðq;p; tÞ ¼1

2p�h

Z 1

�1q� y

2jq̂ðtÞjqþ y

2

D Eeipy=�h dy: ð9Þ

This function is always real, and is analogous to the classical phasespace density that solves the classical Liouville equation. TheWigner function is not a true probability density, however, as itcan assume negative values.

For quantum mechanical systems, the uncertainty principlecomplicates the concept of trajectory, as a system cannot haveboth a definite position xðtÞ and velocity _xðtÞ. In phase space, thestate of a system cannot be localized beyond the limitDqDp ¼ �h=2. Despite this, formulations of quantum mechanics interms of trajectories can be constructed. The path integral formu-lation expresses the propagator hx; tj expð�ibHðt � toÞ=�hÞjxo; toi interms of a sum over all paths connecting the points ðx; tÞ andðxo; toÞ in space–time [12]. Classical Lagrangian mechanics thenemerges from the asymptotic stationary phase limit of the pathintegral expression. A quantum formulation built explicitly on con-figuration space trajectories is given by so-called Bohmian dynam-ics [13–16], where classical-like trajectories evolve under theinfluence of a force derived from the sum of the classical potentialUðxÞ and a quantum potential UQ ðx; tÞ that depends on the wavefunction of the system:

UQ ðx; tÞ ¼ ��h2

2mr2wðx; tÞwðx; tÞ : ð10Þ

Treating quantum systems with classical or classical-like trajecto-ries has been pursued since the early days of quantum theory asboth a method for solving problems and as a tool for interpretation.In chemical physics, this remains an active area of research, wherequantum and semiclassical methods are developed that are basedon using classical molecular dynamics to solve the correspondingquantum equations of motion [17–21,13].

In a recent series of papers, we proposed and developed a tra-jectory-based method for solving the quantum mechanical equa-tions of motion for the Wigner function [22–26]. The quantumLiouville equation is solved numerically by representing it—onaverage—as a positive continuous distribution function approxi-mated by an ensemble of phase space trajectories. Equations ofmotion for these trajectories are derived from the exact equationof motion of the Wigner function. The nonclassical nonlocality ofquantum mechanics leads to a breakdown of the statistical inde-

pendence of the trajectories in the ensemble. The motion of thequantum trajectories are entangled with each other through inter-actions that model quantum nonlocality, and the ensemble mustbe propagated as a unified whole. The method can be used to sim-ulate the dynamics of model quantum system with good accuracy.In addition, it provides insight into the nonlocal nature of quantumdynamics and a classical mechanical perspective on the nonclassi-cal phenomena underlying quantum processes.

The nonlocality of quantum mechanics leads to interactions be-tween the trajectories in the method, and this provides an appeal-ing perspective on the uncertainty principle and quantumentanglement as manifested in phase space representations. How-ever, this effect is more general than one due to the nonclassicalcomponents of quantum mechanics. Any evolution equations inphase space that incorporate additional terms beyond the Poissonbracket structure of Eq. (5) will lead to nonlocal interactions be-tween ensemble members. Indeed, we illustrated these effectsand proposed a corresponding numerical methodology in the con-text of diffusion equations in phase space, such as Eq. (7), in Ref.[24].

In this paper, we describe a general approach to solving evolu-tion equations for probability densities using interacting trajectoryensembles. Assuming the existence of a positive definite (probabi-listic) description of the state of the system, we derive generalequations of motion for the trajectories in the kinematic space(e.g., configuration or phase space). The equations of motion forthe ensemble members depend, in general, on both external forcesand on the probability density itself. This second part then leads tointeraction terms between the ensemble members.

The organization of the rest of the paper is as follows. In Section2, we describe the general formalism for the case of distributions inone dimension. The equations of motion for a continuous distribu-tion are derived, and a variational approach is implemented to de-rive the equations for a finite sampling of the underlyingdistribution by a trajectory ensemble. In Section 3, the approachis applied to modeling a solution of the one- and two-dimensionalSmoluchowski equations describing diffusion under the influenceof time-independent and time-dependent potentials. The resultsare compared with the results of conventional Brownian dynamicssimulations. In Section 4, we discuss trajectory representation ofevolution equations in higher dimensions related to a gauge-likefreedom in the choice of the resulting vector field. The generalprinciples are illustrated with several examples. Finally, a sum-mary and discussion are given in Section 5.

2. Formalism

We start with a general evolution equation in one space coordi-nate x and time t (the generalization to higher dimensions is givenbelow):

@q@t¼ bLq: ð11Þ

Here, qðx; tÞ is a positive probability distribution and bL is a linearoperator. The solution qðx; tÞ can be expressed trivially as the con-volution of itself with a delta function:

qðx; tÞ ¼Z

dðx� nÞqðn; tÞdn: ð12Þ

We now assume that the solution at time t can be written in theform

qðx; tÞ ¼Z

dðx� Xðn; tÞÞqðn; 0Þdn; ð13Þ

where dðx� Xðn; tÞÞ is a propagator that evolves initial density frompoint n at t ¼ 0 to point x at time t. Xðn; tÞ satisfies the initial condi-

22 P. Hogan et al. / Chemical Physics 370 (2010) 20–28

tion Xðn; 0Þ ¼ n. This form preserves the normalization of thedensity:Z

qðx; tÞdx ¼Z

qðx;0Þdx: ð14Þ

We take Eq. (13) as our ansatz, recognizing that the equations ofmotion for the function Xðn; tÞ will be complicated, depending ingeneral on the entire history of the evolution of qðx; tÞ.

We now derive these equations of motion. From Eq. (13), thetime derivative of the density becomes

@qðx; tÞ@t

¼ �Z

_Xðn; tÞd0ðx� Xðn; tÞÞqðn; 0Þdn; ð15Þ

where d0ðx� XÞ ¼ ð@=@xÞdðx� XÞ. The derivative with respect to xacting on the delta function can come out from under the integral,giving

@qðx; tÞ@t

¼ � @

@x

Z_Xðn; tÞdðx� Xðn; tÞÞqðn;0Þdn: ð16Þ

The quantity _Xðn; tÞ is the velocity of the point Xðn; tÞ, which is thepoint that evolves from n as time increases from 0 to t. We can thuswrite _Xðn; tÞ ¼ _XðXðn; tÞÞ � _XðX; tÞ. (Note that with this definition ofnotation we do not mean to imply that X ¼ n. Rather, we express thevector field as a function of the current position X and time t, ratherthan as a function of the initial position n and t.) The delta functionallows us to write _XðX; tÞ ¼ _Xðx; tÞ and the integral then yields

@qðx; tÞ@t

¼ � @

@x½ _Xðx; tÞqðx; tÞ�; ð17Þ

or in terms of the current jðx; tÞ ¼ _Xðx; tÞqðx; tÞ,

@qðx; tÞ@t

þ @jðx; tÞ@x

¼ 0: ð18Þ

This is just the equation of continuity. It allows us to cast the evo-lution equation in the form

� @

@x½ _Xðx; tÞqðx; tÞ� ¼ bLqðx; tÞ: ð19Þ

The vector field as a function of x at time t then becomes, byintegration

_Xðx; tÞ � Vðx; tÞ ¼ � 1qðx; tÞ

Z xbLqðy; tÞdy: ð20Þ

These equations of motion are then integrated to propagate thedensity using Eq. (13).

As a check on our results, consider an infinitesimal time propa-gation of the density. Under the evolution equation, an infinitesi-mal propagation is

qðx; t þ �Þ ’ qðx; tÞ þ �bLqðx; tÞ; ð21Þ

which becomes, in terms of our ansatz,

qðx; t þ �Þ ’Z

dðx� n� � _Xðn;0ÞÞqðn; tÞdn; ð22Þ

where we have used the short time expression

Xðn; t þ �Þ ’ nþ � _Xðn; tÞ ð23Þ

for the trajectory evolution. Expanding the delta function gives

dðx� n� � _Xðn; tÞÞ ’ dðx� nÞ � � _Xðn; tÞd0ðx� nÞ; ð24Þ

where d0ðx� nÞ ¼ ð@=@xÞdðx� nÞ. Inserting this gives

qðx; t þ �Þ ’ qðx; tÞ � �Z

_Xðn; tÞd0ðx� nÞqðn; tÞdn: ð25Þ

Then, using

_Xðn; tÞ ¼ � 1qðn; tÞ

Z nbLqðy; tÞdy; ð26Þ

we obtain

qðx; t þ �Þ ’ qðx; tÞ þ �Z Z nbLqðy; tÞdy� �

d0ðx� nÞdn: ð27Þ

We now note that ð@=@xÞdðx� nÞ ¼ �ð@=@nÞdðx� nÞ, and integrateby parts with respect to n. This gives

qðx; t þ �Þ ’ qðx; tÞ þ �Z bLqðn; tÞdðx� nÞdn: ð28Þ

Finally, performing the integral reproduces Eq. (21), as it should.Numerical implementation involves sampling the density with

a finite number of points,

qðx; tÞ ¼ 1N

XN

j¼1

dðx� xjðtÞÞ; ð29Þ

and then propagating the points using

_xjðtÞ ¼ VðxjðtÞ; tÞ; ð30Þ

where Vðx; tÞ is given by Eq. (20). The function q and its derivativesmust be estimated at each point xj from the evolving ensemble.

In practice, a finite ensemble of N trajectories is employed torepresent the evolving probability distribution. In addition,smoothing functions /ðxÞ must be used for interpolating the solu-tion between ensemble members, allowing derivatives to be com-puted. Here we assume for simplicity the same smoothing functionfor each member of the ensemble; this can be generalized innumerical implementation.

We now derive the equations of motion for a finite smoothedensemble using a variational approach, and investigate its equiva-lence to the analysis given above in the continuum limit. We takean ansatz for the density in terms of an ensemble of trajectories:

qðx; tÞ ¼ 1N

XN

j¼1

/ðx� xjðtÞÞ; ð31Þ

where /ðxÞ is the smoothing function (in practice, a Gaussian).Inserting Eq. (31) into the evolution equation, Eq. (11), gives

@q@t¼ � 1

N

XN

j¼1

_xjðtÞ/0ðx� xjðtÞÞ; ð32Þ

where /0ðxÞ ¼ d/ðxÞ=dx. We now define an error function�ð _x1; _x2; . . . ; _xNÞ as

� ¼Z

@q@t� bLq� �2

dx: ð33Þ

This becomes

�ð _x1; _x2; . . . ; _xNÞ ¼Z XN

j¼1

_xj/0ðx� xjÞ þ bL/ðx� xjÞ

" #2

dx: ð34Þ

This error is to be minimized with respect to the velocities _xk. Thisgives

@�@ _xk¼ 2

Z/0ðx� xkÞ

XN

j¼1

_xj/0ðx� xjÞ þ bL/ðx� xjÞ

" #dx ¼ 0: ð35Þ

Thus, a system of linear equations for the velocities _xk results:

XN

j¼1

ðDkj _xj þKkjÞ ¼ 0; ð36Þ

where

P. Hogan et al. / Chemical Physics 370 (2010) 20–28 23

Dkj ¼Z

/0ðx� xkÞ/0ðx� xjÞdx; ð37Þ

Kkj ¼Z

/0ðx� xkÞbL/ðx� xjÞdx: ð38Þ

Defining the vector k with components

kk ¼XN

j¼1

Kkj; ð39Þ

we can write Eq. (36) as

D _x ¼ �k; ð40Þ

which gives the formal solution for _x as

_x ¼ �D�1k: ð41Þ

It is instructive to derive these quantities explicitly for a simplecase: the diffusion equation in one dimension. Here, the operator bLis given by

bL ¼ D@2

@x2 : ð42Þ

We take the smoothing function to be a Gaussian: /ðxÞ ¼ expð�ax2Þ.Then, the elements of the matrices D and K can be computed ana-lytically. We find

Dkj ¼ a

ffiffiffiffiffiffip2a

r½1� aðxk � xjÞ2�e�

12aðxk�xjÞ2 ; ð43Þ

Kkj ¼ �Da2

ffiffiffiffiffiffip2a

r½3ðxk � xjÞ � aðxk � xjÞ3�e�

12aðxk�xjÞ2 : ð44Þ

We can compare the predictions of this variational approachwith the results obtained previously by taking the continuum limitof the former. The first term of Eq. (36) becomesX

j

Dkj _xj !Z

qðn; tÞZ

/0ðy� xÞ/0ðy� nÞdy� �

_xðn; tÞdn; ð45Þ

where xj ! x and the sum over j becomes an integral over n with theweighting factor of the instantaneous density qðn; tÞ. We can furtherlet the smoothing functions /ðx� yÞ ! dðx� yÞ, yieldingX

j

Dkj _xj !Z

qðn; tÞZ

d0ðy� xÞd0ðy� nÞdy� �

_xðn; tÞdn: ð46Þ

Noting that

d0ðx� nÞ ¼ @

@xdðx� nÞ ¼ � @

@ndðx� nÞ; ð47Þ

we can integrate by parts to findZqðn; tÞ

Zd0ðy� xÞd0ðy� nÞdy

� �_xðn; tÞdn

¼ZZ

@

@n½ _xðn; tÞqðn; tÞ�

� �d0ðy� xÞdðy� nÞdydn ð48Þ

and evaluating the integral over n givesXj

Dkj _xj !Z

@

@y½ _xðy; tÞqðy; tÞ�

� �d0ðy� xÞdy ð49Þ

or, following a second integration by parts, we obtain finally,Xj

Dkj _xj ! �@2

@x2 ½ _xðx; tÞqðx; tÞ�: ð50Þ

For self-adjoint operators bL, a similar analysis yieldsXj

Kkj ! �@

@x½bLqðx; tÞ�: ð51Þ

Putting the pieces together yields

@2

@x2 ½ _xðxÞqðxÞ� ¼ �@

@x½bLqðxÞ�: ð52Þ

This is the derivative with respect to x of the continuity equationform of the evolution equation.

� @jðx; tÞ@x

¼ � @

@x½ _xðx; tÞqðx; tÞ� ¼ bLqðx; tÞ: ð53Þ

Integrating Eq. (52) with respect to x twice (and setting integrationconstants equal to zero) then gives the same result as the equationsof motion derivation:

_xðx; tÞ ¼ � 1qðx; tÞ

Z xbLqðy; tÞdy: ð54Þ

The one-dimensional analysis given above can be generalized tomultidimensional systems. Considering an N dimensional spacespanned by Cartesian coordinates x ¼ ðx1; x2; . . . ; xNÞ, we have thecontinuity equation

@qðx; tÞ@t

þr � jðx; tÞ ¼ 0 ð55Þ

and the relation between the vector field and the evolutionequation

�r � ½ _Xðx; tÞqðx; tÞ� ¼ bLqðx; tÞ: ð56Þ

In the multidimensional case, there is considerable freedom in theintegration of this partial differential equation. In general, we candecompose the operator bL into N terms, as

bL ¼XN

j¼1

k̂j ð57Þ

(some of the terms can be chosen to equal zero). Each of these termscan be paired with one term of the divergence r � ½ _Xðx; tÞqðx; tÞ� onthe left side of Eq. (56), yielding a set of N partial differentialequations:

@

@xjð _Xjðx; tÞqðx; tÞÞ ¼ �k̂jqðx; tÞ ð58Þ

for j ¼ 1;2; . . . ;N. The components of the vector field as functions ofx at time t then become, by integration,

_Xjðx; tÞ � Vjðx; tÞ ¼ �1

qðx; tÞ

Z xj

k̂jqðx1; x2; . . . ; yj; . . . ; xN; tÞdyj ð59Þ

ðj ¼ 1;2; . . . ;NÞ.There is great freedom in the partition given in Eq. (57). For a

separable operator bL ¼PjbLjðxjÞ, it is natural (but not necessary)

to associate each k̂j ¼ bLjðxjÞ. Alternatively, each k̂j can be assignedto a fraction of bL itself:

k̂j ¼ ajbL; ð60Þ

where the sumP

jaj ¼ 1. Many other decompositions can becontemplated.

3. Numerical examples

We now illustrate the general formalism described above byshowing the results of numerical solution of evolution equationsusing the interacting trajectory method, and compare the resultswith the standard method of Brownian dynamics. We treat evolu-tion equations of the form

@qðx; tÞ@t

¼ D@

@x1

kBT@Uðx; tÞ@x

þ @

@x

� �qðx; tÞ ð61Þ

24 P. Hogan et al. / Chemical Physics 370 (2010) 20–28

in one dimension, or

@qðx; tÞ@t

¼ Dr � 1kBTrUðx; tÞ þ r

� �qðx; tÞ ð62Þ

in multiple dimensions, where D ¼ kBT=mc. These correspond todiffusion in overdamped systems, as described by the Smoluchow-ski equation [3].

For the one-dimensional case, the interacting trajectory vectorfield becomes

_Xðx; tÞ ¼ � DkBT

@Uðx; tÞ@x

� Dqðx; tÞ

@qðx; tÞ@x

: ð63Þ

In higher dimensions this generalizes to

_Xðx; tÞ ¼ � DkBTrUðx; tÞ � D

qðx; tÞrqðx; tÞ: ð64Þ

The corresponding Langevin equation describing the Brownian mo-tion of the system is

_xðtÞ ¼ � DkBT

@Uðx; tÞ@x

þ 1mc

RðtÞ; ð65Þ

where RðtÞ is a random force. The generalization to higher dimen-sions is

_xðtÞ ¼ � DkBTrUðx; tÞ þ 1

mcRðtÞ: ð66Þ

We compare numerical solutions of the equations of motion for theinteracting trajectory ensembles with an ensemble of trajectoriesevolving under the stochastic Langevin equation. We consider twoone-dimensional potentials and a two-dimensional system.

The first one-dimensional model consists of a linear plus cosinepotential, given by

UðxÞ ¼ 1� 0:15x� cosð0:6xÞ: ð67Þ

The evolving density qðx; tÞ is approximated by an ensemble ofN trajectories fxjðtÞg ðj ¼ 1;2; . . . ;NÞ and smoothed using

qðx; tÞ ¼ 1N

XN

j¼1

/ðx� xjðtÞÞ ð68Þ

with the smoothing functions /ðxÞ ¼ ð1=ffiffiffiffiffiffiffi2pp

hÞ expð�x2=2h2Þ. Thewidths h of the smoothing functions are given byh ¼ ½4=ðNðdþ 2ÞÞ�1=ðdþ4Þ, where d ¼ 1 is the dimension of the system[27]. We take m ¼ 1, c ¼ 1, and kBT ¼ 3:0 in our simulations. Boththe interacting trajectory and Brownian dynamics simulations areperformed using 1000 trajectories with initial conditions sampledfrom a Gaussian distribution centered at x ¼ 0 with a standard devi-ation r ¼ 1. The interacting trajectory equations of motion aresolved using a Runge–Kutta RK4 algorithm [28] with timestep ofDt ¼ 0:02. The stochastic Langevin equation is integrated usingthe method of Ermak, as described in Ref. [29].

The results of the simulations are shown in Fig. 1. Four snap-shots of the evolving density qðx; tÞ for the interacting trajectorymethod are shown as solid curves. These are compared with theBrownian dynamics simulation, shown as dashed curves. The tworesults are nearly superimposable. Also depicted in the figure isthe potential UðxÞ. The system starts initially ðt ¼ 0Þ in one wellof the metastable system. The system undergoes diffusion and di-rected transport from left to right, down the energy stairway, astime increases.

The second one-dimensional model is a time-dependent saw-tooth potential Uðx; tÞ ¼ uðxÞgðtÞ modeling the operation of aBrownian ratchet [30,31]. The spatial part is periodic in space withperiod L : uðxþ LÞ ¼ uðxÞ. Between x ¼ 0 and x ¼ L it is given by

uðxÞ ¼aðx=xSÞ; 0 < x < xS;

að1� ðx� xSÞ=xLÞ; xS < x < L;

�ð69Þ

where xS þ xL ¼ L. In our calculations xS ¼ 2 and xL ¼ 10. The func-tion gðt þ sÞ ¼ gðtÞ switches periodically between 1 and 0. The per-iod s is taken to be 13.5, and during ð0; sÞ the function is given by

gðtÞ ¼1; 0 < t < 10:0;0; 10:0 < t < 13:5:

�ð70Þ

The barrier height a ¼ 8kBT , and we take kBT ¼ 2:0. Other parame-ters are the same as for Fig. 1.

The results of this model are shown in Fig. 2. Initially, the sys-tem is localized in one minimum of the sawtooth potential. Att ¼ 10:0, the potential switches to zero, and the system undergoesdiffusion. At t ¼ 13:5, the potential turns back on, and its asym-metric shape causes a net transport of the ensemble from left toright. The convention for the plot is the same as in Fig. 1, and,again, close agreement between interacting trajectory and Brown-ian dynamics results is observed.

We now consider a two-dimensional model, constructed byadding a harmonic term in y to the one-dimensional linear plus co-sine system. The frequency of the harmonic oscillator depends onx.

Uðx; yÞ ¼ 12ð1:15� 0:03xÞy2 þ ð1� 0:15x� cosð0:6xÞÞ: ð71Þ

The same parameters and method are employed to solve the equa-tions, but with N ¼ 2000 trajectories for the interacting ensemblemethod. We find that to get smooth results, 5000 trajectories are re-quired for the Brownian simulations.

The two-dimensional results are shown in Fig. 3. Net motiondown the energy stairway, from left to right, is observed, as inthe one-dimensional case of Fig. 1. In addition, spreading in the ydirection is observed, consistent with the weakening of the har-monic restoring force in y as x increases. Excellent agreement be-tween the two methods is again obtained.

4. Gauge freedom in multidimensional systems

We now return to the issue of ambiguity in defining a trajectoryrepresentation for evolution equations in higher dimensions. Thefreedom associated with decomposing the equations of motion isreminiscent of the freedom associated with the choice of gaugein electromagnetic theory [32]. In particular, we can transformthe vector field Vðx; tÞ governing the trajectory representation bya gauge-like transformation

Vðx; tÞ ! V0ðx; tÞ ¼ Vðx; tÞ þWðx; tÞ; ð72Þ

where Wðx; tÞ can be any vector field that satisfies the condition

r � ðWðx; tÞqðx; tÞÞ ¼ 0: ð73Þ

Thus, there are infinitely many trajectory representations of a givenevolution equation, all related by transformations of the form Eqs.(72) and (73).

We illustrate this general principle with a concrete example,consisting of isotropic diffusion in two dimensions. The operatorbL is then given by

bL ¼ D@2

@x2 þ@2

@y2

!: ð74Þ

We decompose the operator using its separability: k̂x ¼ D@2=@x2

and k̂y ¼ D@2=@y2. Eq. (20) then gives the vector field

a b

c d

Fig. 1. Comparison of interacting trajectory ensemble simulation and Brownian dynamics simulation for the linear plus cosine potential, Eq. (67). Four snapshots of theevolving probability distributions are shown. The interacting trajectory method is shown as solid curves. These are compared with the Brownian dynamics simulation, shownas dashed curves. (a) t ¼ 0. (b) t ¼ 25. (c) t ¼ 50. (d) t ¼ 75.

a b

c d

Fig. 2. Comparison of interacting trajectory ensemble simulation and Brownian dynamics simulation for the time-dependent Brownian ratchet potential. Four snapshots ofthe evolving probability distributions are shown. The interacting trajectory method is shown as a solid curve, and compared to the Brownian dynamics simulation, shown as adashed curve. (a) t ¼ 0. (b) t ¼ 25. (c) t ¼ 50. (d) t ¼ 75.

P. Hogan et al. / Chemical Physics 370 (2010) 20–28 25

_Xðx; y; tÞ ¼ � Dqðx; y; tÞ

@qðx; y; tÞ@x

; ð75Þ

_Yðx; y; tÞ ¼ � Dqðx; y; tÞ

@qðx; y; tÞ@y

: ð76Þ

A gauge-like transformation can be performed on this vector fieldby adding a two-dimensional vector ðWx;WyÞ that obeysr � ðWqÞ ¼ 0. Many such vectors are possible; here we consider afamily of vectors given by

Wxðx; y; tÞ ¼d

qðx; y; tÞ@qðx; y; tÞ

@y; ð77Þ

Wyðx; y; tÞ ¼ �d

qðx; y; tÞ@qðx; y; tÞ

@x; ð78Þ

where d is a variable parameter with the same units as the diffusionconstant D. It is easy to check that the divergence of Wq is indeedzero.

In Fig. 4, we show a comparison of two interacting ensemblesimulations. The first, represented by solid lines in the contourplots, is identical to the results of Fig. 3, while the dashed contoursshow the results of including gauge terms in the vector field. Vec-tor fields of the form given in Eqs. (77) and (78) are employed, withd ¼ 1. All other parameters are as before. It is apparent that theevolving functions are virtually identical, although analysis of indi-vidual trajectories in the ensembles show their motion is quite dif-ferent between the two simulations.

As a further example of gauge freedom in trajectory methods,we consider standard Hamiltonian dynamics. Here, the vector fieldin phase space is simply that given by Hamilton’s equations, and is

a b

c d

Fig. 3. Comparison of interacting trajectory ensemble simulation and Brownian dynamics simulation for the two-dimensional potential, Eq. (71). Four snapshots of theevolving probability distributions are shown. The interacting trajectory method is shown as a solid contour lines, and compared to the Brownian dynamics simulation, shownas dashed contours. (a) t ¼ 0. (b) t ¼ 25. (c) t ¼ 50. (d) t ¼ 75.

26 P. Hogan et al. / Chemical Physics 370 (2010) 20–28

independent of the density qðq;p; tÞ. Still, there are infinite possi-ble vector fields which yield the same evolution of q.

The classical Liouville equation is

@q@t¼ bLq ¼ fH;qg; ð79Þ

where qðq;p; tÞ is a density in phase space x ¼ ðq;pÞ;Hðq;pÞ is theHamiltonian, and fA;Bg is the Poisson bracket of A and B:

fA; Bg ¼XN

k¼1

@A@qk

@B@pk� @B@qk

@A@pk¼ rA � J � rB; ð80Þ

where J is the symplectic matrix:

J ¼0 1�1 0

� �ð81Þ

and r is the gradient in phase space:

r ¼@@q@@p

!: ð82Þ

The Liouville equation then becomes

@q@t¼ rH � J � rq: ð83Þ

We can write the Liouville equation as a continuity equation

@q@tþr � j ¼ 0; ð84Þ

where j ¼ _xq. For classical Hamiltonian dynamics the flow in phasespace is incompressible: ðr � _xÞ ¼ 0. In this case the Liouvilleequation becomes

@q@t¼ �ð _x � rÞq: ð85Þ

By noting that JT ¼ �J, we can then identify the equation for thevector field _x as

_x ¼ J � rH; ð86Þ

which are just the usual Hamilton’s equations:

_q ¼ @H@p

;

_p ¼ � @H@q

: ð87Þ

We now consider a gauge-like transformation, of the form

_x! _xþw; ð88Þ

where the invariance of the evolution equation requires

r � ðwqÞ ¼ 0: ð89Þ

If we further require the vector field to itself have a zero divergence,we have the condition

r �w ¼ 0: ð90Þ

Together, these give the gauge condition

w � rq ¼ 0: ð91Þ

The result makes sense, physically: the modification to the vectorfield must be everywhere orthogonal to the direction of change ofq if the transformation is to leave the evolution of q invariant.

We now consider a simple but important example: an invariantdistribution given as a function of the Hamiltonian, q ¼ f ðHÞ. Then,

a b

c d

Fig. 4. Comparison of two interacting trajectory ensemble simulations for the two-dimensional system with potential Uðx; yÞ given by Eq. (71). Four snapshots of the evolvingprobability distributions are shown. The interacting trajectory method is shown as a solid contour lines, and compared to an interacting trajectory simulation modified byaddition of the gauge-like terms to the vector field, given by Eqs. (77) and (78) with d ¼ 1, shown as dashed contours. (a) t ¼ 0. (b) t ¼ 25. (c) t ¼ 50. (d) t ¼ 75.

P. Hogan et al. / Chemical Physics 370 (2010) 20–28 27

w � rq ¼ ðw � rHÞf 0ðHÞ ¼ 0 ð92Þ

or

w � rH ¼ 0: ð93Þ

One solution to this is

w ¼ �J � rH: ð94Þ

This transforms the vector field as follows:

_x! 0: ð95Þ

The result again makes sense: if we sample an invariant distributionwith trajectories and then do not let them move, the distributiondoes not change. In general, we can consider the family of transfor-mations wðaÞ ¼ aJ � rH, where a is an adjustable parameter. Thistransformation yields Hamilton-like equations that differ from theusual ones in that the rate of change of the phase space coordinatescan be tuned by a. For the special case of a ¼ �1, we have the caseof ‘‘time standing still”, as described above, although all values of ayield the correct invariant behavior for q.

We now briefly consider the implications for trajectory-basedapproaches to quantum mechanics. The quantum Liouville equa-tion in the Wigner representation is given by [8–11]

@qW

@t¼ � p

m@qW

@qþ V 0ðqÞ @qW

@p� �h2

24V 000ðqÞ @

3qW

@p3 þ � � � : ð96Þ

For simplicity, we will ignore the higher order terms in what fol-lows. Following our previous work [22–26], we identify the righthand side as (the negative of) the divergence of a quantum fluxwritten in a classical form jW ¼ _xqW , so

@qW

@tþr � jW ¼ 0 ð97Þ

giving

@

@qð _qqWÞ ¼

@

@qpm

qW þ hqqW

� ;

@

@pð _pqWÞ ¼

@

@pð�V 0ðqÞqW þ hpqWÞ; ð98Þ

which defines the ‘‘quantum vector field” h ¼ ðhq; hpÞ. We then con-sider the equation of motion condition

r � ðhqWÞ ¼�h2

24V 000ðqÞ @

3qW

@p3 : ð99Þ

If we want incompressible phase space flow, we should also imposethe divergence condition

r � h ¼ 0: ð100Þ

If the incompressibility condition is ignored and the choice hq ¼ 0,the equation of motion condition can be integrated to yield

hp ¼�h2

24V 000ðqÞ 1

qW

@2qW

@p2 : ð101Þ

This leads to the method described in Refs. [22–26]. Other choicesare possible, however. For instance, one can choose hp ¼ 0, whichthen yields

hq ¼�h2

241

qW

Z q

V 000ðq0Þ @3qW ðq0;pÞ@p3 dq0: ð102Þ

This defines an alternative (and untested) quantum trajectorymethod. Many other divisions of the quantum vector field between

28 P. Hogan et al. / Chemical Physics 370 (2010) 20–28

its q and p components are possible, which all lead to the samequantum Liouville equation.

It would be of interest to consider general solutions to thisproblem which also incorporate the incompressibility conditionr � h ¼ 0.

5. Summary

In this paper, we have described a general formal method for solv-ing evolution equations for probability distributions using trajectoryensembles. For systems described by a positive definite distributionfunction, we derived the general equations of motion for the trajec-tories. We showed that the vector field describing the time rate ofchange of the trajectory ensemble members depends, in general,on both external forces and on the probability density itself. Thedependence of the equations of motion on the probability densitylead to interactions between the ensemble members and a loss oftheir statistical independence. The formalism was illustrated by anumber of numerical examples consisting of diffusion on potentialenergy surfaces. Excellent agreement between the numerical imple-mentation of the interacting trajectory formalism and conventionalBrownian dynamics was obtained. For multidimensional systems, agauge-like freedom exists in the choice of the underlying vector field,which leaves the evolution of the probability density invariant. Weillustrated this point with several examples.

When taken as a mathematical approach to solving partial dif-ferential equations using trajectory ensembles, we have shownthat no unique set of trajectories result. Rather, there are infinitepossibilities that are related by families of gauge transformations.Thus, trajectories lose their realistic interpretation, and becomejust constructs in the solution of the underlying partial differentialequation for the probability distribution. This observation has rel-evance to more philosophical analyses of the foundations andinterpretation of quantum mechanics, the correspondence princi-ple, hidden variable theories, and other contexts where particletrajectories are postulated in quantum mechanical systems.

From a practical standpoint, such transformations may be use-ful in molecular dynamics or stochastic sampling of statisticalmechanical ensembles. There may be particular gauge choices cor-responding to modified dynamics that are more efficient in prac-tice than standard classical mechanics or Langevin dynamics.This will be investigated in future publications.

Acknowledgements

The authors gratefully recognize the many contributions madeby Eli Pollak to the field of theoretical chemistry, and dedicate thispaper to him. This work was supported by the National ScienceFoundation.

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