Rare Events, the Thermodynamic Action and the Continuous-Time Limit
JJournal of Statistical Physics manuscript No. (will be inserted by the editor)
P.J. Malsom · F.J. Pinski
Rare Events, the Thermodynamic Action and theContinuous-Time Limit the date of receipt and acceptance should be inserted later
Abstract
We consider diffusion-like paths that are ex-plored by a particle moving via a conservative force whilebeing in thermal equilibrium with its surroundings. Toprobe rare transitions, we use the Onsager-Machlup (OM)functional as a path probability distribution function fordouble-ended paths that are constrained to start andstop at predesignated points after a fixed time. We ex-plore the continuous-time limit where the OM functionalhas been commonly regularized by using the Ito-Girsanovchange of measure. When used as a path measure, theIto-Girsanov expression generates an ensemble of double-ended paths that are unphysical. We expose the underly-ing reasons why this continuous-time limit does not, andcannot, generate a thermodynamic ensemble of paths.Furthermore, we show that the concept of the Most Prob-able Path and the Thermodynamic action are incompat-ible with such measures for discrete or continuous timediffusion processes.
Keywords
Brownian Dynamics · Pinned Diffusion · Thermodynamic Fluctuations
PACS · · Much of today’s work in the study of diffusion processesis grounded in an expression for the probability of a suc-cession of states of a spontaneously fluctuating thermo-dynamic system. This expression, famously reported inthe 1953 article of Onsager and Machlup [30], has becomeknown as the Onsager-Machlup (OM) functional and hasso far withstood the test of time. One commentary evenstated that the results are “incapable of improvement ei-ther in form or in their mode of derivation” [28]. Here, as
P.J. Malsom · F.J. PinskiDepartment of Physics, University of Cincinnati, Cincinnati,Ohio 45221, USATel.: +1 (513) 556-0523Fax: +1 (513) 556-3425E-mail: [email protected] in the original article, we represent the fluctuations bywhite noise, spatially and temporally uncorrelated, andwhose amplitude is given by the fluctuation-dissipationtheorem. The underlying motion is then expressed interms of Brownian dynamics.For the purposes of this paper, we divide rare eventsinto two groups: those consistent with thermodynamicsand those that are not. As an example of the latter,what we call extremely rare events, consider the verysmall probability that all the molecules in a room mightmigrate to one corner. Even though this is allowed bystatistical mechanics, if we observed such an event, wewould have witnessed a violation of thermodynamics. Weare not, nor were Onsager and Machlup [30], interestedin these extremely rare events. We are interested in rare,events that are consistent with thermodynamics; onesthat are driven by the fluctuations inherent in a thermo-dynamic system.In the 1970s, the original OM functional was ex-tended to the continuous-time limit using a Ito’s lemma,the Girsanov theorem, and the Radon-Nikodyn deriva-tive. The probability measure, which we will refer toas the Ito-Girsanov measure, was written for this limit,and has generated considerable attention [2, 6, 15, 16,18–21, 23, 24, 35, 36, 40, 41]. This limiting procedure hasbeen accepted as a method that can be used to look atrare events by constructing probabilities for pinned dif-fusion paths [5, 14, 15, 22, 26, 32, 39]. An extension of thislimiting procedure, proposed by Graham [15], and laterEyink [10], stressed the generalization of a “least-action”principle to describe particle motions, which then leadsto the notion of the Thermodynamic action.In this paper, we show that this Ito-Girsanov mea-sure, the continuous-time limit of the OM function, is notthe appropriate probability measure for exploring ther-modynamically driven transitions. Furthermore, we showthat the probabilities that arise from such “thermody-namic actions” are independent of the details of the par-ticle motions as the probabilities are a result of the noisethat originates in the thermal bath. Thus there is no“action” to minimize. We explain how the Ito-Girsanov a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec functional is unphysical in that it is inconsistent withthermodynamics, and our direct sampling confirms this.In addition, for thermodynamic significant events, thepath-probability measure is flat, that is, it attains thesame value for any event that is consistent with ther-modynamics. The point that is missed in the long listof works, cited above, is that the Ito-Girsanov expres-sion only provides the change of measure relative to thefree Brownian Bridge. Nowhere has it been proven thatit can be used to give the correct relative probability ofany two paths. On the contrary, here we show that theIto-Girsanov measure is not a path-probability measure,it is only an indicator of how much a free-Brownian so-lution differs from a solution for a nonzero force. Throughout this paper, we will consider a particle in con-tact with a heat reservoir at a temperature (cid:15) . It is mov-ing under the influence of a potential V ( x ) with the forcebeing F ( x ) = − V (cid:48) ( x ). Note that although the equationsare written for the one-dimensional case for clarity, theformalism can easily be extended to higher dimensionsand for a collection of particles.The equation of motion for Brownian dynamics isgiven by the Stochastic Differential Equation (SDE): dx = F ( x ) dt + √ (cid:15) dW t (1)where dW t is the standard Wiener process that repre-sents the (uncorrelated) Gaussian noise. Using a discretetime step, ∆t , one typically uses the Euler-Maruyama al-gorithm [25] as an approximate method for propagatingthe position as a function of time. In particular, x i +1 = x i + F ( x i ) ∆t + √ (cid:15) ∆t ξ i where ξ i is a Gaussian random variate with mean zeroand unit variance. Successive application (N times) ofthis equation produces a sequence of positions { x i } whichis called a path. Onsager and Machlup [30] used the un-derlying thermal fluctuations to write the Gaussian pathprobability, P p ∝ Π i exp( − ξ i ), in terms of the pathvariables themselves, namely, − ln P p = C + ∆t (cid:15) N (cid:88) i =1 (cid:16) x i +1 − x i ∆t − F ( x i ) (cid:17) , (2)where C is a constant which is unimportant for this pa-per. This equation defines what is sometimes called thethermodynamic action and other times, the OM func-tional. In the continuous-time limit, using Ito calculusand the Girsanov theorem, the Radon-Nikodym deriva-tive is used to express the change in the measure [29].We define the Ito-Girsanov measure to be d P p d Q p = exp (cid:32) − (cid:15) (cid:16) V ( x T ) − V ( x ) + (cid:90) T dt G ( x t ) (cid:17)(cid:33) (3) where T is the duration of the path, P p is the continuous-time limit of P p , Q p is the measure associated with freeBrownian motion, and the function G ( x ) is defined as G ( x ) = F ( x ) · F ( x ) − (cid:15) V (cid:48)(cid:48) ( x ). One of the uses of the OM functional is to incorporateit into a scheme to sample paths that are constrained atboth ends. The aim is to efficiently generate an ensembleof paths that include a transition over an energy barrier.When the barrier is large compared to the typical ther-mal energy, the transition is a rare event. We wish toexplore barrier hopping that is consistent with thermo-dynamics, where the driving noise reflects the fluctuatingrandom effects that originate in a thermal bath and thusare independent of the particle’s position.The quandary is that for a very simple one-dimensionalexample, the generated paths quickly become unphysi-cal when using the Ito-Girsanov form, P p , for the pathprobability. Long paths, generated with small time steps,are expected to be consistent with equilibrium thermo-dynamics. First we address the question of what happensaway from the continuous-time limit and show that it ispossible to use a method which generates a collectionof paths that are consistent with the Boltzmann distri-bution. Then we show that when we use Ito-Girsanovexpression, given by Eq. 3, we obtain results that areunphysical.We have constructed a one-dimensional potential tohighlight one of the key elements in this problem, namely,the role of entropy. We explore whether entropic effectsare properly included by forming a potential having twodegenerate wells with different widths. In particular, con-sider the potential (plotted in Fig. 1) V ( x ) = 2 − (8 − x ) (2 + 5 x ) (4)which has two degenerate wells with a barrier of unityat the origin. A narrow (quadratic) well is on the leftand the wide well is on the right. By making the wellsdegenerate, we eliminate the large (exponential) depen-dence due to energy differences. And we accentuate theentropic effects by making one well much wider than theother.To understand how the time discretization affects thenumerical solution to the SDE we used both Eq. 2 andthe following form of the path probability P sp , − ln P sp = C + ∆t (cid:15) N (cid:88) i =1 (cid:32) (cid:12)(cid:12)(cid:12) x i +1 − x i ∆t − F ( x i ) (cid:12)(cid:12)(cid:12) − J i (cid:33) (5)where J i = (cid:15)∆t ln (cid:16) − ∆t F (cid:48) ( x i ) (cid:17) , x i is the midpointof x i and x i +1 . This expression, P sp , is related to theStratonovich representation of the diffusion process (seeElber and Shalloway [8]). As noted by Van Kampen [38], - Position x P o t en t i a l V Fig. 1
A plot of the double-well potential used in this pa-per (see Eq. 4). A narrow (quadratic) well, on the left, isseparated from the broad well on the right by an unit energybarrier. for the problem described by Eq. 1, the limits of boththe Ito and Stratonovich discretizations are equivalent.In the continuous-time limit J i ⇒ (cid:15) F (cid:48) ( x i ) which pro-vides the Laplacian term in the definition of G , givingthe same limit as seen in the Ito-Girsanov expression inEq. 3.In sampling P p and P sp , we generated a sequence ofpaths at a temperature (cid:15) = 0 .
25, constrained to start inthe narrow well and to end in the broad well and used aHMC implementation using the “Implicit Algorithm” inBeskos et al. [4].We use the Heaviside function Θ to define the func-tion B ( s ) to be the fraction of the path that is containedin the broad well, namely, B ( s ) = 1 T (cid:90) T dt Θ ( x ( s ) t ) ≈ N (cid:88) i Θ ( x ( s ) i )where the sampling index is denoted as s , and the cor-responding path is { x ( s ) i } . As can be seen in Fig. 2, using either of the path prob-abilities given by a discrete sum, we find an ensemble ofpaths, which are concentrated in the wide well. Thesepaths are constrained to undergo at least one transi-tion; we find only one transition in the majority of paths.We also find paths which undergo multiple transitions.Using either expression, on average, the percent of thetime spent in the broad well is approximately 80% whichis much closer to the equilibrium value of 90%. This isthe type of behavior that one expects for paths consis-tent with the Boltzmann distribution. Away from thecontinuous-time limit, we found a behavior that is closeto the physical result. In the next section, we show thatthese are in stark contrast to calculations that sample thepath probability expression given by the Ito-Girsanov ex-pression (Eq. 3).
Now we turn the results found when sampling from thepath probability given by the Ito-Girsanov expression, P p (Eq. 3). Results generated using the Ito-Girsanovexpression has been questioned before. For example inAdib’s paper [1], the origins of the poor performancewere not understood. The potential, described above andused in this work, was designed to highlight entropic ef-fect on transition paths. With degenerate wells, entropyenhances the time the particle will spent in the widewell. However, from the form of P p , the curvature of thenarrow basin would seem to enhance the probability ofpaths that spend a long time there, see the early pa-per of Weiss and H¨affner [40]. Indeed the Most ProbablePath [31] (MPP) would be highly concentrated in thenarrow well.We used the Ito-Girsanov formula with the methoddeveloped by Beskos et al [3], referred to here as path-space Hybrid Monte-Carlo (psHMC), using a tempera-ture (cid:15) = 0 .
25, to generate a sequence of paths constrainedto start in the narrow well and to end in the broad well.In Fig. 2, we plot B ( s ) using the described procedure(red curve), showing that the paths quickly become un-physical in that the particle spends the vast majority ofthe time in the left, thin well. Such paths are inconsistentwith the equilibrium thermodynamical distribution. Al-though only one sampling set is presented here, we notethat this effect is robust in that similar results are ob-tained for a range of parameters. Fig. 2
The fraction of the path in the broad well is shown asa function of an arbitrary sampling index, s . These are resultsfor three calculation. We started with the same input pathwhere B (0) ≈ .
6, and had a path length of T = N ∆t = 150,and a time step along the path of ∆t = 0 . Fig. 3
Results for the calculation using the RWM algorithmstarting from a randomly generated Brownian Bridge. As afunction of iteration number, a plot of the fraction of thepath which resides in the Broad Well (black line) and theequilibrium value (orange dotted line).
We now turn to a simpler sampling method, the RandomWalk Metropolis (RWM) algorithm, to sample P p , whichproceeds as follows: First, create a Brownian path thatstarts and ends at the origin. A new Brownian Bridgeis generated and then combined with the current path(with the sum of the squares of the mixing coefficients be-ing unity). A proposed path is generated from the Brow-nian Bridge by shifting the starting point to be x − = 0 . x + = 1 .
6. The acceptance orrejection of this new path is based on P p (Eq. 3). Thisprocedure was repeated thousands of times and gener-ated the results displayed in Fig. 3. As shown in thisfigure, the starting Brownian Bridge evolves to a paththat is concentrated in the broad well. The fraction ofthe time that the path spends in the positive half planebegins at a value below 0 .
40 and settles to a value largerthan 0 .
95. This result differs from the sampling of theIto-Girsanov expression shown above. However, if onestarts with a path that is highly concentrated (94%) inthe narrow well instead of a pinned Brownian path, onefinds that the path does not evolve significantly. Look-ing at Eq. 3, one finds that the integral of G over theinitial path is a large negative number. The mixing pa-rameter in this case had to be reduced by an order ofmagnitude otherwise any proposed path was rejected bythe Metropolis step. Clearly, the integral of G derives itslarge negative value from the path being concentratedin the narrow well. Looking at the sine transform of thepath, one would see that the low frequency part of thespectrum looks more like that of an Ornstein-Uhlenbeck(OU) process [37] rather than that of free Brownian mo-tion. Thus the proposed moves generated in the RWMmethod are ill-suited for probing the fluctuations aroundthis sharp probability maximum. To recap the results presented above: we found that bysampling pinned paths we were able to generate differentresults. On the one hand, we generated an ensemble ofpaths that were inconsistent with the Boltzmann distri-bution by using the Ito-Girsanov expression and 1) usingpsHMC algorithm or 2) with the RWM starting from avery particular initial path. On the other hand, we gen-erated paths that concentrate in the wide well, using aform of the OM functional with an HMC algorithm 3)with an Ito discretization or 4) with a Stratonovich dis-cretization, away from the continuum limit, and 5) usingthe Ito-Girsanov expression with the RWM starting froma Brownian Bridge.The key to understanding numerics behind these re-sults is to examine the “cross term” in Eq. 2. This termcan be written as − x i +1 − x i ∆t · F ( x i ) = x i +1 − x i ∆t · (cid:16) F ( x i +1 ) − F ( x i ) (cid:17) − x i +1 − x i ∆t · (cid:16) F ( x i +1 ) + F ( x i ) (cid:17) The last term above is an approximation to the change-in-potential term in Eq. 3 (as it telescopes under sum-mation). The previous term is approximately x i +1 − x i ∆t · (cid:16) F ( x i +1 ) − F ( x i ) (cid:17) ≈ ∆x ∆t · F (cid:48) ( x ) (6)where x is the midpoint between x i and x i +1 . A furthersimplification can be done, but only if the time incrementis small and the noise and positions are uncorrelated.For sufficiently small ∆t , one can average over severaltime steps. Doing so, we arrive at an expression thatapproaches the Ito-Girsanov expression (Eq. 3), namely, ∆x ∆t · F (cid:48) ( x ) → − (cid:15) V (cid:48)(cid:48) ( x ) . (7)Remember that the noise, as embodied in ∆x , mustbe uncorrelated to the positions, as required in the un-derlying SDE (Eq. 1) when probing thermodynamicallyconsistent motions. It is important to note that this re-quirement is satisfied (by construction) when the RWMmethod is used, but only if the correlation is absent inthe initial path. This means that the some of the symp-toms of the unphysical probability measure will remainhidden as the RWM method lacks the flexibility to fullysample the measure.In the psHMC algorithm the implied noise historyand the particle position are intertwined as one searchesfor the paths that have the larger probabilities as indi-cated by the Ito-Girsanov expression. During the search,correlations are introduced into the low end of the fre-quency spectrum while the path evolves under the (de-terministic) Hamiltonian flow. High frequencies are cor-rectly handled as paths generated by both methods have quadratic variations that are close to ideal: (cid:80) ∆x ≈ (cid:15) T . The implication of such low-frequency correlationis that the thermal bath is no longer independent of thesystem and thus the fluctuations are no longer thermo-dynamic, and entropy is generated [9].The resulting paths (not shown here) using the psHMCmethod look like a noisy version of the so-called mostprobable path [31], MPP. Recall that the MPPs weregenerated by finding the probability maximum impliedby the Ito-Girsanov expression. Thus we conclude thatthe unphysical nature of the MPPs found previously [31]is a direct result of using this unphysical measure. RWMproduces paths that are only mixture of Brownian pathswhich evidently lack the flexibility to probe the regionsthat the measure indicates are of high probability.Understanding this cross term (Eq. 6) helps to ex-plain why we see less of a problem with calculations foran OU process. When F (cid:48) is a constant, the cross term isaveraged correctly if the grid spacing is fine enough sothat the Quadratic Variation is close to ideal.We note that our findings are not touched by theIto vs. Stratonovich controversy. In the problem consid-ered here, the noise and the “diffusion coefficient” arespatially homogeneous. As pointed out by Van Kam-pen [38] both discretization schemes lead to the samecorrect limit. Looking at Eq. 5, it is clear that in thelimit as ∆t →
0, we recover the Ito-Girsanov functional,Eq. 3, if we regularize the expression. It is the regu-larization step that hides an assumption. When usingthe Radon-Nikodym derivative, the Ito-Girsanov mea-sure cannot be the probability measure because it hasmeaning only when using free Brownian Bridges and notfor a general path.
We now turn to a more general discussion of the ideaof an MPP in the OM formalism. We first examine therelative probability of paths. As in any continuous dis-tribution, while the probability of any one path is zero,the relative probability of two paths is well-defined. Inthe case of free Brownian motion, for sufficiently longpaths of the same duration, the relative probability ofany two paths is unity. But this is also true when con-sidering Brownian dynamics with any force. In all cases,the path probability is governed by the Gaussian noise, P p ∝ Π i exp( − ξ i ), where the set { ξ i } is independentof the force and thus independent of the position of theparticle. The same argument applies to any finite repre-sentation of the path, and thus holds as the size of thetime increment becomes infinitesimally small.To illustrate the above point, consider the following“thought” experiment. For a conservative force considersampling the Boltzmann distribution using the Brown-ian dynamics as expressed by Eq. 1. Take the startingpoint, x , to be arbitrary and integrate the SDE over - Position ( x ) N o r m a li z ed C oun t Fig. 4
Distribution of endpoints of the set of trajectorieswith equal probability. Here the endpoints of 472,640 differ-ent trajectories are shown. The blue line is the normalizedBoltzmann distribution, exp ( − V /(cid:15) ) /Z , for the temperature (cid:15) = 0 .
25, and potential given by Eq. 4. a fixed time, T, that is long compared to any barrierhopping time. Using a non-zero but small time step, ∆t ,one uses N r = T /∆t
Gaussian random variates. Keepingthe same set of random numbers, but simply scramblingthe order, redo the integration. This provides N r ! pathseach with identical probabilities. For large enough T andsmall enough ∆t , the set of endpoints, x ( T ) should bedistributed in a manner that is close to Boltzmann. Hereit is important to recognize that it is not the path proba-bility that creates the distribution, as all paths are iden-tically probable. Rather it is the path density that drivesthe correct distribution of the endpoints. We have par-tially preformed this experiment (calculation) for the po-tential defined in Eq. 4. We chose N r = 2 million randomnumbers, Gaussian distributed with mean zero and vari-ance one. Out of the possible N r ! paths we followed over400,000 of them using the Euler-Maruyama algorithm,with ∆t = 0 . G . It is clearly incorrect to use theIto-Girsanov expression in this manner. And in particu- lar, this is the immediate origin of the numerical resultsdisplayed above. The question is how do we understand our results in lightof the above information? To accomplish this, we look atthe ramifications for an Ornstein-Uhlenbeck (OU) pro-cess [37] with the force being F OU = − γ x . The Ito-Girsanov change of measure indicates that the probabil-ity distributions of two diffusions that differ only in theirdrift term are mutually absolutely continuous. Exact so-lutions are known for both the free Brownian motion andthe OU process. For the OU process, the frequency ( ν )spectrum is finite at the origin with a 1 /ν tail. The fre-quency spectrum of the free particle motion falls off as1 /ν after diverging at the origin. For a long enough pathlength, every realization of the Brownian Bridge has thissame spectrum. Since linear combinations of such realiza-tions are still (free) Brownian Bridges, they too have thesame spectrum. Thus each Brownian Bridge and everylinear combination of Brownian Bridges have a frequencyspectrum that differs from that of an OU process; theyare not solutions to the OU SDE. This holds true forother forces; in general, in the limit as the path lengthbecomes infinite, free Brownian Bridges are not solutionsto the SDE (Eq. 1) with nonzero drift. This then explainswhy the RWM sampling method is increasingly sluggishas the path length increases.With this in mind, we now examine the Kullback-Leibler divergence D KL ( P || Q ) between the measures fortwo diffusions P and Q that differ in their drift terms.If we take Q to represent the measure of free Brown-ian Bridges then the expectation of the Ito-Girsanov ex-pression can be identified as the Kullback-Leibler diver-gence since both measures have the same “normaliza-tion” terms as inferred from the discussion in the pre-vious section. This then explains why the Ito-Girsanovchange of measure should be viewed as an indicator ofthe differences between the solutions of two SDEs thatdiffer only in their drift term and is not related to theprobability of paths of one or the other diffusions. The Onsager-Machlup functional is based on Browniandynamics and provides a way of understanding the dou-ble ended path sampling problem [15]. It had been ac-cepted that in the continuous-time limit, the Onsager-Machlup functional could be replaced by the Ito-Girsanovchange of measure (Eq. 3) as a way of handling the infini-ties inherent in such a limit. We have shown here in threeways that this is inappropriate. First, direct samplinggives unphysical results. Second, in the long path limit,any linear combination of free Brownian paths cannot be a solution to the OU process nor to diffusions with morecomplicated forces. Third, interpreting the Ito-Girsanovchange of measure as a probability distribution favorssome paths over others even though paths of the sameduration must have the same probability. We determinedthat the Ito-Girsanov change of measure is simply an in-dicator of how inaccurately the free Brownian paths rep-resent solutions to the diffusion process with a nonzerodrift. Thus using the Ito-Girsanov change of measure asa probability distribution is not thermodynamically cor-rect.The sampling of paths is a complicated numericalprocedure interlaced with sophisticated mathematics. Insuch problems, it can be difficult to separate concep-tual errors from numerical artifacts. It was previouslyobserved that the MPPs [31] for many potentials seemedto be unphysical, which should have been taken as an in-dication that something was amiss. Now we understandwhy. The Ito-Girsanov expression is not a path probabil-ity measure and thus the minimizers of the Ito-Girsanovexpression are not related to physical paths.The particular potential, used here, was constructedto highlight entropic effects on transition paths. Usingit, we were able to pinpoint a common misconception.We have found similar behaviour with other potentials,such as two quadratic wells with differing widths (con-nected with a third-order spline). Also, in two dimen-sions, we studied diffusion in the potential V ( x, y ) =( x + y − exp( − x − y +1) ) , which has an almostdegenerate channel. There we found unphysical resultsfor a temperature (cid:15) = 0 .
10; these results are consistentwith the MPP in that the path has a tendency to con-centrate near the point (0 ,
1) where the channel walls aresteepest, rather than in the broad wells. But such detailsare unimportant in that they again demonstrate similarbehavior as found in the one-dimensional case that wehave reported here. The results for the simple double-well potential shown in this paper clearly illustrate thatthe Ito-Girsanov expression does not correctly describeentropic effects when used as a basis for sampling paths.The findings in this article affect a wide body ofwork. As mentioned above, the theory for entropy pro-duction in nonequilibrium thermodynamics, such as thework of Speck et al [34], should be re-examined. In-deed it would seem that even an encyclopedia entry [27]needs to be modified. For probing the folding of pro-teins, the recent work of Fujisaki, et al [13] suffered fromusing the unphysical form of the Onsager-Machlup func-tional. It is unclear how to interpret their results. Otherworks [7,11,17,42] have to be reevaluated in light of thesenew results.The results reported here also have ramifications forsampling general measures in infinite dimensions. Thesamples or paths in infinite dimension must be constructedto probe the high probability regions as defined by themeasure. Algorithms based on Brownian Bridges have avery limited flexibility in that the noise history is un- correlated by construction. Only in HMC methods, highprobability regions are explored due to the Hamiltonianflow which allows the high probability, highly correlated,modes to be explored. The limited sophistication of thesampling algorithms employed in the past has hid someof the results exposed in this article.The path integrals that appear in diffusion look sim-ilar to those that appear in Quantum Mechanics. Thenatural question is what is impact of the current work onFeynman Path Integrals [12]. The short answer is none.Firstly the paths in Quantum Mechanics are smooth,while Brownian paths are almost nowhere differentiable.Secondly diffusion is a local process while solutions to theSchr¨odinger equation are not. The time evolution of thewave function is most easily found by using an eigenfunc-tion expansion; the eigenfunctions are dependent on thepotential everywhere in space. Diffusion is described bya Markovian process and white noise; local in space andtime. Thirdly, in the path integral formulation of Quan-tum Mechanics, the path distribution is peaked aroundthe classical path, a smooth curve that is defined by theclassical action. As we discussed above, each physical(thermodynamically-consistent) diffusion bridge has thesame probability. The most important observation in thisregard is that the field-theoretic methods, such as in thereview by Smith [33], developed for Feynman integralsare not transferable to classical diffusion.
Acknowledgements
We wish to especially thank Robin Ball,Andrew Stuart, Hendrik Weber, Gideon Simpson, and Flo-rian Theil for many lengthy conversations. We acknowledgethe use of computing resources provided by the Open ScienceGrid.
References
1. Adib, A.B.: Stochastic actions for diffusive dynamics:Reweighting, sampling, and minimization. The Journalof Physical Chemistry B (19), 5910–5916 (2008)2. Bach, A., D¨urr, D., Stawicki, B.: Functionals of pathsof a diffusion process and the onsager-machlup function.Zeitschrift f¨ur Physik B Condensed Matter (2), 191–193 (1977)3. Beskos, A., Pinski, F., Sanz-Serna, J., Stuart, A.: Hybridmonte carlo on hilbert spaces. Stochastic Processes andtheir Applications (10), 2201–2230 (2011)4. Beskos, A., Roberts, G., Stuart, A., Voss, J.: Mcmc meth-ods for diffusion bridges. Stochastics and Dynamics (03), 319–350 (2008)5. Dai Pra, P., Pavon, M.: Variational path-integral rep-resentations for the density of a diffusion process.Stochastics: An International Journal of Probability andStochastic Processes (4), 205–226 (1989)6. D¨urr, D., Bach, A.: The Onsager-Machlup function asLagrangian for the most probable path of a diffusion pro-cess. Comm. Math. Phys. (2), 153 –170 (1978)7. E, W., Ren, W., Vanden-Eijnden, E.: Minimum actionmethod for the study of rare events. Communications onPure and Applied Mathematics (5), 637–656 (2004).DOI 10.1002/cpa.200058. Elber, R., Shalloway, D.: Temperature dependent re-action coordinates. The Journal of Chemical Physics (13), 5539–5545 (2000). DOI http://dx.doi.org/10.1063/1.4811319. Esposito, M., Lindenberg, K., den Broeck, C.V.: Entropyproduction as correlation between system and reservoir.New Journal of Physics (1), 013,013 (2010). URL http://stacks.iop.org/1367-2630/12/i=1/a=013013
10. Eyink, G.L.: Action principle in statistical dynamics.Progress of Theoretical Physics Supplement , 77–86(1998)11. Faccioli, P., Sega, M., Pederiva, F., Orland, H.: Dominantpathways in protein folding. Phys. Rev. Lett. , 108,101(2006)12. Feynman, R., Hibbs, A.: Quantum mechanics and pathintegrals. International series in pure and applied physics.McGraw-Hill (1965). URL https://books.google.com/books?id=14ApAQAAMAAJ
13. Fujisaki, H., Shiga, M., Kidera, A.: Onsager-Machlupaction-based path sampling and its combination withreplica exchange for diffusive and multiple pathways. TheJournal of Chemical Physics (13), 134101 (2010).DOI http://dx.doi.org/10.1063/1.337280214. Goovaerts, M., De Schepper, A., Decamps, M.: Closed-form approximations for diffusion densities: a path in-tegral approach. Journal of computational and appliedmathematics , 337–364 (2004)15. Graham, R.: Path integral formulation of general diffu-sion processes. Zeitschrift fur Physik B Condensed Mat-ter (3), 281–290 (1977)16. Graham, R., Riste, T.: Fluctuations, instabilities andphase transitions. Plenum, New York p. 215 (1975)17. Hartmann, C., Sch¨utte, C.: Efficient rare event simula-tion by optimal nonequilibrium forcing. Journal of Sta-tistical Mechanics: Theory and Experiment (11),P11,004 (2012)18. Horsthemke, W., Bach, A.: Onsager-machlup function forone dimensional nonlinear diffusion processes. Zeitschriftf¨ur Physik B Condensed Matter (2), 189–192 (1975)19. Hunt, K.L.C., Ross, J.: Path integral solutions of stochas-tic equations for nonlinear irreversible processes: Theuniqueness of the thermodynamic lagrangian. The Jour-nal of Chemical Physics (2), 976–984 (1981). DOIhttp://dx.doi.org/10.1063/1.44209820. Ikeda, N., Watanabe, S.: The Onsager-Machlup functionsfor diffusion processes. In: K. Matthes (ed.) StochasticDifferential Equations and Diffusion Processes, pp. 510–510. WILEY-VCH Verlag (1986). DOI 10.1002/bimj.4710280425. URL http://dx.doi.org/10.1002/bimj.4710280425
21. Ito, H.: Probabilistic construction of lagrangian of diffu-sion process and its application. Progress of TheoreticalPhysics (3), 725–741 (1978)22. Ito, H.: Optimal gaussian solutions of nonlinear stochas-tic partial differential equations. Journal of statisticalphysics (5-6), 653–671 (1984)23. Langouche, F., Roekaerts, D., Tirapegui, E.: Shortderivation of feynman lagrangian for general diffusionprocesses. Journal of Physics A: Mathematical and Gen-eral (2), 449 (1980)24. Lavenda, B.: Thermodynamic criteria governing the sta-bility of fluctuating paths in the limit of small thermalfluctuations: critical paths in the limit of small thermalfluctuations: critical paths and temporal bifurcations.Journal of Physics A: Mathematical and General (17),3353 (1984)25. Maruyama, G.: Continuous markov processes andstochastic equations. Rendiconti del Circolo Matematicodi Palermo (1), 48–90 (1955). DOI 10.1007/BF0284602826. Matsumoto, H., Yor, M., et al.: Exponential functionalsof brownian motion, ii: Some related diffusion processes.Probability Surveys , 348–384 (2005)27. McKane, A.J.: Stochastic processes. In: Encyclope-dia of Complexity and Systems Science, pp. 8766–8783.Springer (2009)28. McKean, H.P.: In: The Collected Works of Lars Onsager:With Commentary, Lars Onsager and P.C. Hemmer, pp.769–771. World Scientific, Singapore (1998)29. Øksendal, B.: Stochastic Differential Equations. Uni-versitext. Springer Berlin Heidelberg, Berlin, Heidel-berg (2003). URL http://link.springer.com/10.1007/978-3-642-14394-6
30. Onsager, L., Machlup, S.: Fluctuations and irreversibleprocesses. Phys. Rev. (6), 1505 (1953)31. Pinski, F.J., Stuart, A.M.: Transition paths in moleculesat finite temperature. The Journal of Chemical Physics (18), 184104 (2010). DOI 10.1063/1.3391160. URL http://link.aip.org/link/?JCP/132/184104/1
32. Ren, W., Vanden-Eijnden, E., et al.: Minimum actionmethod for the study of rare events. Communications onpure and applied mathematics (5), 637–656 (2004)33. Smith, E.: Large-deviation principles, stochastic effectiveactions, path entropies, and the structure and meaningof thermodynamic descriptions. Reports on Progress inPhysics (4), 046,601 (2011). URL http://stacks.iop.org/0034-4885/74/i=4/a=046601
34. Speck, T., Engel, A., Seifert, U.: The large deviation func-tion for entropy production: the optimal trajectory andthe role of fluctuations. Journal of Statistical Mechanics:Theory and Experiment (12), P12,001 (2012)35. Stratonovich, R.: On the probability functional of diffu-sion processes. Selected Trans. in Math. Stat. Prob ,273–286 (1971)36. Tisza, L., Manning, I.: Fluctuations and irreversible ther-modynamics. Physical Review (6), 1695 (1957)37. Uhlenbeck, G.E., Ornstein, L.S.: On the theory of theBrownian motion. Phys. Rev. , 823–841 (1930). DOI10.1103/PhysRev.36.82338. Van Kampen, N.: Itˆo versus stratonovich. Journal ofStatistical Physics (1), 175–187 (1981)39. Watabe, M., Shibata, F.: Path integral and brownian mo-tion. Journal of the Physical Society of Japan (6),1905–1908 (1990)40. Weiss, U., H¨affner, W.: The uses of instantons for diffu-sion in bistable potentials. Functional Integration: The-ory and Applications p. 311 (1980)41. Yasue, K.: The role of the onsager–machlup lagrangianin the theory of stationary diffusion process. Journal ofMathematical Physics20