Abstract
A path integral formalism for non-equilibrium systems is proposed based on a manifold of quasi-equilibrium densities. A generalized Boltzmann principle is used to weight manifold paths with the exponential of minus the information discrepancy of a particular manifold path with respect to full Liouvillean evolution. The likelihood of a manifold member at a particular time is termed a consistency distribution and is analogous to a quantum wavefunction. The Lagrangian here is of modified generalized Onsager-Machlup form. For large times and long slow timescales the thermodynamics is of Öttinger form. The proposed path integral has connections with those occuring in the quantum theory of a particle in an external electromagnetic field. It is however entirely of a Wiener form and so practical to compute. Finally it is shown that providing certain reasonable conditions are met then there exists a unique steady-state consistency distribution.
Similar content being viewed by others
Notes
With the proviso that \(p\) is always Gaussian as was originally assumed by Onsager and Machlup.
More general invariants than energy of the dynamical system may also be considered.
Note that this Lagrangian is quite distinct from that applying to the original Hamiltonian dynamics. It can in some sense be regarded as a slow variable Lagrangian for the system since \(\lambda \) specifies the slow variable expectation values via a Legendre transform.
The energy function for TBH is simply the sum of the squares of the spectral mode amplitudes meaning the Gibbs measure is a Gaussian with uncorrelated modes and equal variances proportional to the conserved energy of the system.
Note that we use the terminology distribution here to avoid confusion with the approximating trial densities. The consistency distribution is a function (or distribution) of the coordinates \(\lambda \) which specify the position within the manifold of trial densities. The densities are defined on the original variables of the Hamiltonian system.
If the trial distribution gives an invariant measure for the system this will not be the case.
Strictly this identification as a projection is precise only in the limit as \(\Delta t\rightarrow 0\).
We use \(M\) to denote the magnetic vector potential to avoid confusion with the resolved variable set \(A\).
This can include both an electric potential and other potentials such as gravitation.
Set by the static space-time Killing vector.
Or Fourier/phase decompositions.
This is strictly a Wick rotated Schrödinger equation i.e. a parabolic PDE of a diffusion-absorbtion type.
Up to a scalar multiple and the addition of a function vanishing almost everywhere with respect to the Lebesgue measure.
References
Amari, S., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs, AMS, Oxford University Press (2000)
Battezzati, M.: Onsager principle for nonlinear mechanical systems modeled by stochastic dissipative equations. Arch. Mech. 64(2), 177–206 (2012)
Ceperley, D.M.: Path integrals in the theory of condensed helium. Rev. Mod. Phys. 67(2), 279 (1995)
Chaichian, M., Demichev, A.: Path integrals in physics. Vol. 1: Stochastic processes and quantum mechanics. IOP, London (2001)
Conway, J.B.: A Course in Functional Analysis. Springer, Berlin (1990)
Cover, T., Thomas, J.: Elements of Information Theory, 2nd edn. Wiley-Interscience, New York (2006)
Darve, E., Solomon, J., Kia, A.: Computing generalized Langevin equations and generalized Fokker-Planck equations. Proc. Natl Acad. Sci. 106(27), 10884–10889 (2009).
Deininghaus, U., Graham, R.: Nonlinear point transformations and covariant interpretation of path integrals. Z. Phys. B Con. Mat. 34(2), 211–219 (1979)
Dekker, H.: On the path integral for diffusion in curved spaces. Phys. A 103(3), 586–596 (1980)
Du, Y.: Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Vol. 1: Maximum Principles and Applications. World Scientific Publishing Company (2006)
Einstein, A.: Theorie der Opaleszenz von homogenen Fluessigkeiten und Fluessigkeitsgemischen in der Naehe des kritischen Zustandes. Ann. Physik 33, 1275 (1910)
Eyink, G.L.: Action principle in nonequilibrium statistical dynamics. Phys. Rev. E 54(4), 3419–3435 (1996)
Feynman, R.P., Hibbs, A.R., Styer, D.F.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)
Gardiner, C.W.: Handbook of Stochastic Methods for Physics. Springer, Chemistry and the Natural Sciences (2004)
Graham, R.: Path integral formulation of general diffusion processes. Z. Phys. B Con. Mat. 26(3), 281–290 (1977)
Haken, H.: Generalized Onsager-Machlup function and classes of path integral solutions of the Fokker-Planck equation and the Master equation. Z. Phys. B Con. Mat. 24(3), 321–326 (1976)
Hanche-Olsen, H., Holden, H.: The Kolmogorov-Riesz compactness theorem. Expositiones Mathematicae 28, 385–394 (2010)
Kleeman, R., Turkington, B.E.: A nonequilibrium statistical model of spectrally truncated Burgers-Hopf dynamics. Commun. Pure Appl. Math. 67(12), 1905–1946 (2014)
Kraichnan, R.H.: Variational method in turbulence theory. Phys. Rev. Lett. 42(19), 1263–1266 (1979)
Landau, L.D., Lifshitz, E.M.: Quantum Mechanics Non-relativistic Theory. Pergamon, London (1965)
Lavenda, B.H.: On the validity of the Onsager-Machlup postulate for nonlinear stochastic processes. Found. Phys. 9(5–6), 405–420 (1979)
Lieb, E.H., Loss, M.: Analysis. American Mathematical Society, Providence (2001)
Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91(6), 1505–1512 (1953)
Öttinger, H.C.: Beyond Equilibrium Thermodynamics. Wiley-Interscience (2005)
Roncadelli, M.: New path integral representation of the quantum mechanical propagator. J. Phys. A-Math. Gen. 25(16), L997 (1992)
Taniguchi, T., Cohen, E.G.D.: Onsager-Machlup theory for nonequilibrium steady states and fluctuation theorems. J. Stat. Phys. 126(1), 1–41 (2007)
Turkington, B.: An optimization principle for deriving nonequilibrium statistical models of hamiltonian dynamics. J. Stat. Phys. 152, 569–597 (2013)
Ventsel, A.D., Freidlin, M.I.: On small random perturbations of dynamical systems. Russ. Math. Surv. 25(1), 1–55 (1970)
Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)
Zubarev, D.N.: Nonequilibrium Statistical Thermodynamics. Plenum Press, New York (1974)
Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, New York (2001)
Acknowledgments
Comprehensive discussions with Bruce Turkington on matters related to the present contribution are very gratefully acknowledged. Useful discussions on related matters over many years with Andy Majda are also acknowledged. This paper is dedicated to my mother Annette.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix A: Some Useful Relations
Define the expectation bracket
for a general function of the state variables and time \(F\). We have now
where we are using the anti-Hermitian nature of \(L\) on the second line. Setting \(F=1\) we obtain immediately that
Now it is easily derived from the definition of \(R\) and the form of the exponential family of distributions that
which when combined with (51) yields
The anti-Hermitian nature of \(L\) also allows us to deduce the following two useful relations (using the summation convention and vector/matrix indices for clarity):
where we are using the fact that \(L\) annihilates \(E\) and (53) for the second last step. Combining (53) and (54) we obtain
In a completely analogous way to (54) we deduce that
and more generally
It is easily shown also that
Appendix B: Section 8 Theorem Proof
We first establish that the operator \(K\) is bounded with respect to the \(L_{1}\) norm. Consider the effect of \(K\) on a distribution \(\psi \) with \(L_{1}\) norm unity:
where on line 4 we have used the fundamental fact derived in section 2 that \(IL_{rev}\ge 0\) while the last line follows after switching variables of integration and using the normalization condition which also holds for \(\exp \left[ -IL_{irr}\right] \).
We further establish that the image of \(K\) is totally bounded which means establishing the additional two properties (48) and (49).
From condition 1. of the Theroem we deduce that there exists a \(\left| \kappa _{0}\right| \) such that
Consider now the bounded region \(\left| \kappa \right| \le \left| \kappa _{0}\right| \). From condition 2. of the Theorem; the region boundedness and the fact that \(\exp \left( -IL_{irr}\right) \) is Gaussian in \(\lambda \), if follows that there exists an \(r(\epsilon )>0\) such that for all \(\left| \kappa \right| \le \left| \kappa _{0}\right| \)
Thus
which establishes (48).
To establish the other required property consider an arbitrary \(\rho _{t}>0\) and all \(\gamma \) with \(\left| \gamma \right| <\rho _{t}\). The triangle inequality plus (48) implies that
Set \(S(\epsilon ,\rho )=R(\frac{\epsilon }{4})+\rho \) and \(V_{\epsilon \rho _{t}}\) the volume of the region \(Z:\left| \lambda \right| \le S(\epsilon ,\rho _{t})\). Let \(\kappa _{0}\) be such that \(\left| \kappa \right| >\left| \kappa _{0}\right| \) implies that
We have
with
an integral transform defined on a bounded domain. The first integral on the RHS of (58) is easily shown using the triangle inequality; the inequality (57) and the non-negativity of the \(IL\) terms to be less than \(\frac{\epsilon }{4}\).
The function \(f_{\kappa _{0}}\) can be shown by standard arguments to be continuous since the integral transform is defined on a bounded domain and the function \(R\) is continuous with respect to the first argument. By the Heine-Cantor theorem it is therefore uniformly continuous on the bounded region \(Z\). It follows that there exists a \(\rho _{U}\) such that for all \(\gamma :\left| \gamma \right| \le \rho _{U}\) and all \(\lambda \in Z\)
and so for such \(\gamma \) the second integral from (58) is also less than \(\frac{\epsilon }{4}.\) Compare now \(\rho _{U}\) and \(\rho _{t}\). If \(\rho _{U}\ge \rho _{t}\) then we can replace \(\rho _{U}\) with \(\rho _{t}\) in the last integral inequality discussed and obtain the required inequality (49) by combining the three inequalities derived from (56) and (58). Conversely if \(\rho _{U}<\rho _{t}\) then \(V_{\epsilon \rho }<V_{\epsilon \rho _{t}}\) Thus inequalities (57) and (59) still hold if we use \(\rho _{U}\) in place of \(\rho _{t}\). Furthermore the newly defined bounded \(Z\) is a subset of the old \(Z\) whence the uniform continuity just discussed holds with the same \(\rho _{U}\) and hence we are done.
Rights and permissions
About this article
Cite this article
Kleeman, R. A Path Integral Formalism for Non-equilibrium Hamiltonian Statistical Systems. J Stat Phys 158, 1271–1297 (2015). https://doi.org/10.1007/s10955-014-1149-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-014-1149-x