A Path Integral Formalism for Non-equilibrium Hamiltonian Statistical Systems

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.

Appendices

Appendix A: Some Useful Relations

Define the expectation bracket

$$\begin{aligned} \left\langle F\right\rangle \equiv \int F\hat{p} \end{aligned}$$

for a general function of the state variables and time $F$. We have now

$$\begin{aligned} \frac{\partial \left\langle F\right\rangle }{\partial t}-\left\langle L(F)\right\rangle&= \left\langle \frac{\partial F}{\partial t}\right\rangle +\int (F\hat{p}_{t}-L(F)\hat{p})\nonumber \\&= \left\langle \frac{\partial F}{\partial t}\right\rangle +\int (F(\partial _{t}+L)\hat{p}\nonumber \\&= \left\langle \frac{\partial F}{\partial t}\right\rangle +\int FR\hat{p}\nonumber \\&= \left\langle \frac{\partial F}{\partial t}+FR\right\rangle \end{aligned}$$

(50)

where we are using the anti-Hermitian nature of $L$ on the second line. Setting $F=1$ we obtain immediately that

$$\begin{aligned} \left\langle R\right\rangle =0 \end{aligned}$$

(51)

Now it is easily derived from the definition of $R$ and the form of the exponential family of distributions that

$$\begin{aligned} R=\dot{\lambda }^{t}(A-a)+\lambda {}^{t}LA \end{aligned}$$

(52)

which when combined with (51) yields

$$\begin{aligned} \lambda ^{t}\left\langle LA\right\rangle =0 \end{aligned}$$

(53)

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):

(54)

where we are using the fact that $L$ annihilates $E$ and (53) for the second last step. Combining (53) and (54) we obtain

$$\begin{aligned} \lambda ^{t}h\lambda =0 \end{aligned}$$

In a completely analogous way to (54) we deduce that

$$\begin{aligned} \left\langle L^{2}(A_{i})\right\rangle =-\lambda _{j}\left\langle LA_{j}LA_{i}\right\rangle \equiv -k_{ij}\lambda _{j}. \end{aligned}$$

and more generally

$$\begin{aligned} \left\langle L^{n}A_{j}\right\rangle =-\lambda _{i}\left\langle LA_{i}L^{n-1}A_{j}\right\rangle \end{aligned}$$

(55)

It is easily shown also that

$$\begin{aligned} \frac{\partial M_{i}}{\partial \lambda _{j}}=\left\langle L(A_{i})\left( A_{j}-a_{j}\right) \right\rangle =h_{ji} \end{aligned}$$

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:

$$\begin{aligned} \left\| K\psi \right\| _{1}&= \int \left| \int R\psi d\kappa \right| d\lambda \\&\le \int \left[ \int \left| R\psi \right| d\kappa \right] d\lambda \\&= \iint N\exp \left[ -IL_{irr}-IL_{rev}\right] \left| \psi \right| d\kappa d\lambda \\&\le \iint N\exp \left[ -IL_{irr}\right] \left| \psi \right| d\kappa d\lambda \\&= \int \left| \psi \right| d\kappa =1 \end{aligned}$$

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

$$\begin{aligned} \exp \left( -IL_{rev}\right) <\frac{\epsilon }{2}\qquad if\;\left| \kappa \right| >\left| \kappa _{0}\right| \end{aligned}$$

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 $\left| \kappa \right| \le \left| \kappa _{0}\right|$

$$\begin{aligned} \int \limits _{\left| \lambda \right| >R(\epsilon )}N\exp \left( -IL_{irr}\right) d\lambda <\frac{\epsilon }{2} \end{aligned}$$

Thus

$$\begin{aligned} \int \limits _{\left| \lambda \right| >r(\epsilon )}\left| K\psi \right| d\lambda =&\int \limits _{\left| \lambda \right| >r(\epsilon )}d\lambda \left| \int d\kappa N\exp \left( -IL_{irr}-IL_{rev}\right) \psi \right| \\ \le&\int \limits _{\left| \lambda \right| >r(\epsilon )}d\lambda \int d\kappa N\exp \left( -IL_{irr}-IL_{rev}\right) \left| \psi \right| \\ \le&\int \limits _{\left| \lambda \right| >r(\epsilon )}d\lambda \int \limits _{\left| \kappa \right| \le \left| \kappa _{0}\right| }d\kappa N\exp \left( -IL_{irr}\right) \left| \psi \right| \\&+\int \limits _{\left| \kappa \right| >\left| \kappa _{0}\right| }d\kappa \exp \left( -IL_{rev}\right) \left| \psi \right| \\ <&\frac{\epsilon }{2}+\frac{\epsilon }{2} \end{aligned}$$

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

$$\begin{aligned} \int \limits _{\left| \lambda \right| >r(\frac{\epsilon }{4})+\rho _{t}}\left| K\psi \left( \lambda +\gamma \right) -K\psi (\lambda )\right| d\lambda&\le \int \limits _{\left| \lambda \right| >r(\frac{\epsilon }{4})+\rho _{t}}\left| K\psi \left( \lambda +\gamma \right) \right| d\lambda \nonumber \\ +\int \limits _{\left| \lambda \right| >r(\frac{\epsilon }{4})+\rho _{t}}\left| K\psi (\lambda )\right| d\lambda&\le \frac{\epsilon }{4}+\frac{\epsilon }{4} \end{aligned}$$

(56)

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

$$\begin{aligned} \exp \left( -IL_{rev}\right) <\frac{\epsilon }{8V_{\epsilon \rho _{t}}} \end{aligned}$$

(57)

We have

$$\begin{aligned}&\int \limits _{\left| \lambda \right| \le S(\epsilon .\rho _{t})}\left| K\psi \left( \lambda +\gamma \right) -K\psi (\lambda )\right| d\lambda \nonumber \\&\quad =\int \limits _{\left| \lambda \right| \le S(\epsilon .\rho _{t})}\int \limits _{\left| \kappa \right| >\left| \kappa _{0}\right| }\exp \left( -IL_{red}\right) N\left| \exp \left( -IL_{irr}(\lambda +\gamma \right) -\exp \left( -IL_{irr}(\lambda \right) \right| \left| \psi \right| d\kappa d\lambda \nonumber \\&\qquad +\int \limits _{\left| \lambda \right| \le S(\epsilon .\rho _{t})}\left| f_{\kappa _{0}}(\lambda +\gamma )-f_{\kappa _{0}}(\lambda )\right| d\lambda \end{aligned}$$

(58)

with

$$\begin{aligned} f_{\kappa _{0}}\left( \lambda \right) \equiv \int \limits _{\left| \kappa \right| \le \left| \kappa _{0}\right| }R(\lambda ,\kappa )\psi (\lambda )d\kappa \end{aligned}$$

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 $\lambda \in Z$

$$\begin{aligned} \left| f_{\kappa _{0}}(\lambda +\gamma )-f_{\kappa _{0}}(\lambda )\right| <\frac{\epsilon }{4V_{\epsilon \rho _{t}}} \end{aligned}$$

(59)

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.