Fock Space and Field Theoretic Description of Nonequilibrium Work Relations

We begin with a model of $N$ classical particles on a $d$-dimensional lattice of size $L^{d}$; the continuum limit will be taken later. The particles interact with the pair potential $V_{ij}$, which can be any function of the separation ${\bf r}_{j}-{\bf r}_{i}$ between sites $i$ and $j$, such as a Lennard-Jones potential, with the only restriction that it is symmetric under interchange of particles, $V_{ij}=V_{ji}$. Additionally, we consider a time-dependent site potential $U_{i}(t)$, which allows for work to be done on the system. The occupancy of site $i$ is denoted $n_{i}$, and the energy of the occupation configuration ${\bf n}=(n_{1},n_{2},\dots)$ is given by

$$E_{\bf n}(t)=\frac{1}{2}\sum_{i,j}(n_{i}-\delta_{ij})V_{ij}n_{j}+\sum_{i}U_{i}(t)n_{i},$$

(3)

where the Kronecker delta function ensures proper counting of the same-site pairs.

These particles are coupled to a thermal reservoir at temperature $T$. Because of the time-dependent potential $U_{i}(t)$, the equilibrium distribution also depends on time $t$ via

$$P^{\rm eq}_{\bf n}(t)=\frac{N!}{{\bf n}!}\,\frac{1}{Z(t)}e^{-\beta E_{\bf n}(t)}.$$

(4)

where $\beta^{-1}=k_{B}T$, $Z(t)$ is the partition function associated with $E_{\bf n}(t)$, and ${\bf n}!\equiv\prod_{i}n_{i}!$. The prefactor $N!/{\bf n}!$ is the multinomial coefficient that accounts for the multiplicity of microstates for configuration ${\bf n}$ under permutation of the particles.

The particles undergo stochastic hops, and the probability $P({\bf n},t)$ of configuration ${\bf n}$ at time $t$ obeys the master equation

$$\frac{d}{dt}P_{\bf n}(t)=\sum_{{\bf m}\neq{\bf n}}\Bigl{[}w_{{\bf n},{\bf m}}(t)P_{\bf m}(t)-w_{{\bf m},{\bf n}}(t)P_{\bf n}(t)\Bigr{]}.$$

(5)

The particle hopping rates $w_{{\bf n},{\bf m}}(t)$ for a transition ${\bf m}\to{\bf n}$ are chosen to obey detailed balance for the instantaneous energy $E_{\bf n}(t)$,

$$\frac{w_{{\bf n},{\bf m}}(t)}{w_{{\bf m},{\bf n}}(t)}=\frac{P^{\rm eq}_{\bf n}(t)}{P^{\rm eq}_{\bf m}(t)}=\frac{{\bf m}!}{{\bf n}!}e^{-[\beta E_{\bf n}(t)-\beta E_{\bf m}(t)]}$$

(6)

and are thus time-dependent.

Work is performed on the system by varying the site potentials $U_{i}(t)$. For configuration ${\bf n}$ at time $t$, the instantaneous rate of work performed on the system is given by

$$\dot{W}=\sum_{i}n_{i}\dot{U}_{i}(t).$$

(7)

This term appears in the nonequilibrium generalization of the first law:

$$\frac{d}{dt}\langle E\rangle=\sum_{\bf n}\biggl{[}\frac{dP_{\bf n}(t)}{dt}E_{\bf n}(t)+P_{\bf n}(t)\sum_{i}n_{i}\dot{U}_{i}(t)\biggr{]}.$$

(8)

The first term gives the rate of change in $\langle E\rangle$ due to the particle hopping dynamics, reflecting the heat flow between the system and reservoir, while the second term is simply the average rate of work performed, $\langle\dot{W}\rangle$.

For sufficiently slow driving, the system remains in thermal equilibrium and the probabilities are given by Eq. (4). In contrast, for fast driving, the distribution $P_{\bf n}(t)$ obtained from Eq. (5) may be driven far from equilibrium.

Bosonic creation and annihilation operators $\hat{a}_{i}^{\dagger}$ and $\hat{a}_{i}$ are introduced at each lattice site and, together with the vacuum state $|0\rangle$, used to create a state $|{\bf n}\rangle=\prod_{i}(\hat{a}_{i}^{\dagger})^{n_{i}}|0\rangle$ corresponding to the configuration ${\bf n}$. Note that this state is normalized as as $\langle{\bf n}|{\bf n}\rangle={\bf n}!$. The full probability distribution of the system at time $t$ can then be represented by the state

$$|\psi(t)\rangle=\sum_{\bf n}P_{\bf n}(t)|{\bf n}\rangle,$$

(9)

which enables writing the master equation as

$$\frac{d}{dt}|\psi(t)\rangle=-\hat{L}(t)|\psi(t)\rangle.$$

(10)

with Liouvillian operator $\hat{L}$. The utility of the Doi representation is that whenever the particle dynamics respects the permutation symmetry of the particles, the operator $\hat{L}$ can be expressed solely in terms of the $\hat{a}_{i}$ and $\hat{a}_{i}^{\dagger}$ operators, with no dependence on the configuration ${\bf n}$. That is, the Liouvillian is determined solely by the processes involved and not dependent on the state of the system. For example, particles undergoing unbiased hops between neighboring lattice sites leads to

$$\hat{L}_{\rm diff}=\Gamma\sum_{\langle ij\rangle}(\hat{a}_{i}^{\dagger}-\hat{a}_{j}^{\dagger})(\hat{a}_{i}-\hat{a}_{j}),$$

(11)

where the sum runs over $i,j$ that are nearest neighbor sites.

In the Doi representation, averages and distributions are extracted from the Fock states by means of a projection state, defined as $\langle{\cal P}|=\langle 0|e^{\sum_{i}\hat{a}_{i}}$, with the property $\langle{\cal P}|\hat{a}_{i}^{\dagger}=\langle{\cal P}|$ for all $i$. Averages are computed via

$$\langle A\rangle\equiv\sum_{\bf n}A_{\bf n}P_{\bf n}(t)=\langle{\cal P}|\hat{A}|\psi(t)\rangle,$$

(12)

where $\hat{A}\equiv\sum_{\bf n}(A_{\bf n}/{\bf n}!)|{\bf n}\rangle\langle{\bf n}|$ is an operator diagonalized by the occupation states $|{\bf n}\rangle$ with eigenvalues $A_{\bf n}$. Normalization ensures $\langle{\cal P}|\psi(t)\rangle=\sum_{\bf n}P_{\bf n}(t)=1$, and probability conservation requires

$$0=\frac{d}{dt}\langle 1\rangle=-\langle{\cal P}|\hat{L}(t)|\psi(t)\rangle$$

(13)

for all $|\psi(t)\rangle$, thus

$$\langle{\cal P}|\hat{L}(t)=0.$$

(14)

This is obeyed, for example, by the diffusion Liouvillian in Eq. (11).

Now we consider the class of Liouvillians that correspond to transition rates that satisfy detailed balance, and for clarity we suppress the explicit time dependence in $\hat{L}$ and the rates. Starting from a state ${\bf m}$, the probability of being in a state ${\bf n}\neq{\bf m}$ a short time $\delta t$ later is, according to the master equation, $P({\bf n},t)=w_{{\bf n},{\bf m}}\delta t$, while the probability of remaining in ${\bf m}$ is $P_{\bf m}(\delta t)=1-\delta t\sum_{{\bf n}^{\prime}\neq{\bf m}}w_{{\bf n}^{\prime},{\bf m}}$. In the Doi representation, the state at time $\delta t$ is $|\psi(\delta t)\rangle=(1-\hat{L}\delta t)|{\bf m}\rangle$. Using $\langle{\bf n}|\psi(t)\rangle={\bf n}!P_{\bf n}(t)$ we obtain for ${\bf n}\neq{\bf m}$

$$\langle{\bf n}|\hat{L}|{\bf m}\rangle=-{\bf n}!\,w_{{\bf n},{\bf m}}$$

(15)

while for ${\bf n}={\bf m}$

$$\langle{\bf n}|\hat{L}|{\bf n}\rangle=-{\bf n}!\sum_{{\bf n}^{\prime}\neq{\bf n}}w_{{\bf n}^{\prime},{\bf n}}.$$

(16)

For rates that obey the detailed balance condition of Eq. (6), we obtain

$$\langle{\bf n}|\hat{L}|{\bf m}\rangle=\langle{\bf m}|\hat{L}|{\bf n}\rangle e^{-\beta E_{\bf n}(t)+\beta E_{\bf m}(t)}.$$

(17)

with $E_{\bf n}$ given by Eq. (3). Now we introduce the hermitian Hamiltonian operator

$$\hat{H}(t)=\frac{1}{2}\sum_{i,j}V_{ij}\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{a}_{i}+\sum_{i}U_{i}(t)\hat{a}_{i}^{\dagger}\hat{a}_{i}$$

(18)

which is diagonalized by the states $|{\bf n}\rangle$ with eigenvalues $E_{\bf n}(t)$. With this energy operator Eq. (17) becomes $\langle{\bf n}|\hat{L}|{\bf m}\rangle=\langle{\bf m}|e^{\beta\hat{H}}\hat{L}e^{-\beta\hat{H}}|{\bf n}\rangle$, which implies

$$\hat{L}^{\dagger}=e^{\beta\hat{H}}\hat{L}e^{-\beta\hat{H}}.$$

(19)

Together Eqs. (14) and (19) imply that necessary and sufficient conditions for detailed balance are that $\hat{L}$ has the form

$$\hat{L}=\hat{Q}e^{\beta\hat{H}}$$

(20)

where $\hat{Q}$ is a hermitian operator that annihilates the projection state: $\hat{Q}|{\cal P}\rangle=0$.

Finally, we discuss specific choices for the Liouvillian. Consider a particle hop from site $i$ to neighboring site $j$, which takes the system from configuration ${\bf m}$ to ${\bf n}$, where $n_{k}=m_{k}-\delta_{ik}+\delta_{jk}$. The detailed balance condition (6) for this hop is

$$\frac{w_{{\bf n},{\bf m}}}{w_{{\bf m},{\bf n}}}=\frac{m_{i}}{n_{j}}e^{-\beta[E_{\bf n}-E_{\bf m}]}.$$

(21)

We restrict consideration to rates that depend on the energies only via the difference $\Delta E\equiv E_{\bf n}-E_{\bf m}$. These have the general form

$$w_{{\bf n},{\bf m}}=\Gamma m_{i}f_{e}(\beta\Delta E)e^{-\beta\Delta E/2},$$

(22)

where $f_{e}(x)$ is an even function.

To construct the Liouvillian corresponding to (22) it is useful to introduce the hermitian operator

$$\hat{\epsilon}_{k}=U_{k}+\sum_{\ell}V_{k\ell}\hat{a}_{\ell}^{\dagger}\hat{a}_{\ell}$$

(23)

with eigenvectors $|{\bf m}\rangle$ and eigenvalues $\epsilon_{k}({\bf m})=U_{k}+\sum_{\ell}V_{k\ell}m_{\ell}$ equal to the energy required to introduce a particle at site $k$ to configuration ${\bf m}$. It is straightforward to show that for the $i\to j$ jump

$$(\hat{\epsilon}_{j}-\hat{\epsilon}_{i})\,\hat{a}_{i}|{\bf m}\rangle=\Delta E\,\hat{a}_{i}|{\bf m}\rangle,$$

(24)

which allows us to write (22) as

$$w_{{\bf n},{\bf m}}=\frac{1}{{\bf n}!}\langle{\bf n}|\Gamma\hat{a}_{j}^{\dagger}f_{e}(\hat{\epsilon}_{i}-\hat{\epsilon}_{j})e^{\beta(\hat{\epsilon}_{i}-\hat{\epsilon}_{j})/2}\hat{a}_{i}|{\bf m}\rangle.$$

(25)

By comparison with (15), we can identify the operator above with the gain term in $\hat{L}$.

For the loss term, we consider the same $i\to j$ hop but starting in ${\bf n}$ and going to ${\bf n}^{\prime}$, where $n_{k}^{\prime}=n_{k}-\delta_{ik}+\delta_{jk}$, and $\Delta E=E_{{\bf n}^{\prime}}-E_{\bf n}$. The appropriate rate is obtained from (16) and (22) to be

$$w_{{\bf n}\to{\bf n}^{\prime}}=\frac{1}{{\bf n}!}\langle{\bf n}|\Gamma\hat{a}_{i}^{\dagger}f_{e}(\hat{\epsilon}_{i}-\hat{\epsilon}_{j})e^{\beta(\hat{\epsilon}_{i}-\hat{\epsilon}_{j})/2}\hat{a}_{i}|{\bf n}\rangle.$$

(26)

Combining (25) and (26) with the analogous terms for a $j\to i$ hop provides the Liouvillian corresponding to (22),

$$\hat{L}=\Gamma\sum_{\langle ij\rangle}(\hat{a}_{i}^{\dagger}-\hat{a}_{j}^{\dagger})f_{e}(\hat{\epsilon}_{i}-\hat{\epsilon}_{j})\Bigl{(}e^{\beta(\hat{\epsilon}_{i}-\hat{\epsilon}_{j})/2}\hat{a}_{i}-e^{\beta(\hat{\epsilon}_{j}-\hat{\epsilon}_{i})/2}\hat{a}_{j}\Bigr{)}.$$

(27)

As advertised, the Liouvillian is purely an expression of the process and has no dependence on the state of the system, i.e., the occupation numbers. The identity $e^{\beta\hat{\epsilon}_{k}}\hat{a}_{k}=e^{-\beta\hat{H}}\hat{a}_{k}e^{\beta\hat{H}}$, which follows from $[\hat{H},\hat{a}_{k}]=-\hat{\epsilon}_{k}\hat{a}_{k}$ and the Hadamard lemma, can be used to show this Liouvillian has the necessary form (20).

Later, when taking the continuum limit, the energy difference due to a hop will be of the order of the lattice spacing $\Delta x$, which will allow us to linearize (27) in $\hat{\epsilon}_{i}-\hat{\epsilon}_{j}$:

$$\hat{L}\simeq\Gamma\sum_{\langle ij\rangle}(\hat{a}_{i}^{\dagger}-\hat{a}_{j}^{\dagger})\biggl{(}\hat{a}_{i}-\hat{a}_{j}+\beta(\hat{\epsilon}_{i}-\hat{\epsilon}_{j})\frac{\hat{a}_{i}+\hat{a}_{j}}{2}\biggr{)},$$

(28)

where we have set $f_{e}(0)=1$ without loss of generality. Note that this linearized Liouvillian is normal ordered.

The Jarzynski and Crooks relations require starting in thermal equilibrium. In the Doi representation the equilibrium state can be written as

$$|\psi_{\text{eq}}\rangle=\sum_{\bf n}P_{\bf n}^{\text{eq}}|{\bf n}\rangle=\frac{1}{Z}e^{-\beta\hat{H}}|\bar{n}_{0}\rangle,$$

(29)

where $\bar{n}_{0}=N/L^{d}$ is the average number of particles per site, and $|\bar{n}_{0}\rangle\equiv e^{\sum_{i}\bar{n}_{0}\hat{a}_{i}^{\dagger}}|0\rangle$. This is obtained from (4) by expressing the multiplicity of the occupation state ${\bf n}$ in terms of a product of Poisson distributions: $N!/{\bf n}!\propto\prod_{i}\bar{n}_{0}^{n_{i}}/n_{i}!$, with the implied constraint $\sum_{i}n_{i}=N$ and the proportionality constant absorbed into $Z$.

The equilibrium state is necessarily a stationary state of the detailed balance Liouvillian, Eq. (20). In the Doi representation this can be seen from

$$\hat{L}|\psi_{\text{eq}}\rangle=Z^{-1}\hat{Q}|\bar{n}_{0}\rangle=0,$$

(30)

where the last equation follows from taking $\hat{a}\to\bar{n}_{0}\hat{a}$ and $\hat{a}^{\dagger}\to\bar{n}_{0}^{-1}\hat{a}^{\dagger}$, which leaves $\hat{Q}$ invariant and takes $|\bar{n}_{0}\rangle\to|{\cal P}\rangle$.

Finally, we note that all of $\hat{L}$, $\hat{H}$, $E$, $Z$, and $|\psi_{\text{eq}}\rangle$ are defined in terms of the instantaneous $U_{i}(t)$, and thus can be time-dependent.

Coherent states are introduced at each lattice site,

$$|{\bm{\phi}}\rangle=e^{-\frac{1}{2}\sum_{i}|\phi_{i}|^{2}+\sum_{i}\phi_{i}\hat{a}_{i}^{\dagger}}|0\rangle$$

(45)

with ${\bm{\phi}}=(\phi_{1},\phi_{2},\dots)$ and complex $\phi_{i}$. These are eigenstates of the annihilation operator, $a_{i}|{\bm{\phi}}\rangle=\phi_{i}|{\bm{\phi}}\rangle$. The identity operator can be expressed with the overcompleteness relation $\bm{1}=\int\prod_{i}\frac{d^{2}\phi_{i}}{\pi}|{\bm{\phi}}\rangle\langle{\bm{\phi}}|$. Time evolution is broken into discrete steps of size $\Delta t$, as shown in (31), and the identity operator is inserted at each time slice with a distinct set of coherent states, leading to evaluation of terms of the form

$$\langle{\bm{\phi}}_{t+\Delta t}|1-\hat{L}_{t}\Delta t|{\bm{\phi}}_{t}\rangle=\langle\bm{\phi}_{t+\Delta t}|\bm{\phi}_{t}\rangle[1-\Delta t{\cal L}(\bm{\phi^{*}}_{t+\Delta t},\bm{\phi}_{t})].$$

(46)

Using the linearized, normal ordered Liouvillian (28) we obtain

$${\cal L}(\bm{\phi^{*}}_{t+\Delta t},\bm{\phi}_{t})=\Gamma\sum_{\langle ij\rangle}(\phi^{*}_{i,t+\Delta t}-\phi^{*}_{j,t+\Delta t})\left(\phi_{i,t}-\phi_{j,t}+\frac{\phi_{i,t}+\phi_{j,t}}{2}\left\{\beta\epsilon_{i}({\bm{\phi}}_{t+\Delta t},{\bm{\phi}}_{t})-\beta\epsilon_{j}({\bm{\phi}}_{t+\Delta t},{\bm{\phi}}_{t})\right\}\right)$$

(47)

with the effective potential

$$\epsilon_{i}({\bm{\phi}}_{t+\Delta t})=U_{i}+\sum_{k}V_{ik}\phi^{*}_{k,t+\Delta t}\phi_{k,t}.$$

(48)

The coherent state overlap in (46) can be written as

$$\langle\bm{\phi}_{t+\Delta t}|\bm{\phi}_{t}\rangle=e^{\frac{1}{2}|\bm{\phi}_{t+\Delta t}|^{2}-\frac{1}{2}|\bm{\phi}_{t}|^{2}-\bar{\bm{\phi}}_{i,t+\Delta t}\cdot(\bm{\phi}_{t+\Delta t}-\bm{\phi}_{t})}$$

(49)

where $|{\bm{\phi}}|^{2}=\sum_{i}|\phi_{i}|^{2}$ and $\bar{\bm{\phi}}_{1}\cdot\bm{\phi}_{2}=\sum_{i}\phi^{*}_{1,i}\phi_{2,i}$. The squared terms cancel between successive time slices.

We take the continuum limit via $\phi_{i,t}\to\phi({\bf x},t)\Delta x^{d}$ and $\phi^{*}_{i,t}\to\bar{\phi}({\bf x},t)$, so $\phi({\bf x})$ has dimensions of density while $\bar{\phi}({\bf x})$ is non-dimensional. The result is an action $S$ containing a “bulk” contribution as well as initial and final contributions at $t=0$ and $t_{f}$, which can be used to compute averages via

$$\langle A\rangle=\int{\cal D}(\bar{\phi},\phi)\,A(\bar{\phi},\phi)e^{-S[\bar{\phi},\phi]}$$

(50)

From Eqs. (46) and (49) we obtain the bulk action

$$S_{B}=\int_{0}^{t_{f}}dt\int d{\bf x}\biggl{[}\bar{\phi}(\partial_{t}-D\nabla^{2})\phi-\gamma\bar{\phi}\nabla\cdot\left(\phi\nabla U+\phi\nabla\int d{\bf y}V({\bf x}-{\bf y})\bar{\phi}({\bf y})\phi({\bf y})\right)\biggr{]}$$

(51)

with diffusion constant $D=\Gamma\Delta x^{2}$ and mobility $\gamma=D/k_{B}T$.

Note that for noninteracting particles, with $V=0$, the action (51) is linear in $\bar{\phi}$, which creates a delta function when integrated over $\bar{\phi}$. The resulting field $\phi$ then obeys

$$\frac{\partial\phi}{\partial t}=D\nabla^{2}\phi+\gamma\nabla\cdot(\phi\nabla U)$$

(52)

which is the Fokker-Planck equation for particles diffusing in a potential $U({\bf x})$.

The final term, consisting of the projection state and part of the coherent state overlap, contributes a $t_{f}$ “boundary” contribution to the action of the form

$$e^{-S_{f}}=e^{\frac{1}{2}|{\bm{\phi}}_{t_{f}}|^{2}}\langle{\cal P}|{\bm{\phi}}\rangle=e^{\sum_{i}\phi_{i,t_{f}}}$$

(53)

from which we obtain the continuum limit

$$S_{f}=-\int d{\bf x}\;\phi({\bf x},t_{f}).$$

(54)

Generally this term is eliminated by performing a Doi shift. Instead, we will retain this term, since it plays an essential role in understanding the time-reversal symmetry.

The initial term also provides a $t=0$ boundary term

$$e^{-S_{i}}=Z(0)^{-1}e^{-\frac{1}{2}|\bm{\phi}_{0}|^{2}}\langle\bm{\phi}_{0}|e^{-\beta\hat{H}_{0}}|\bar{n}_{0}\rangle,$$

(55)

In the case of noninteracting particles, where $\hat{H}=\sum_{j}U_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j}$, this becomes

$$e^{-S_{i}}=Z(0)^{-1}e^{-|\bm{\phi}_{0}|^{2}}\exp\left(\sum_{j}e^{-\beta U_{j,0}}\phi^{*}_{j,0}\bar{n}_{0}\right)$$

(56)

where we used the coherent state identity $\langle\phi_{2}|e^{\lambda\hat{a}^{\dagger}\hat{a}}|\phi_{1}\rangle=\exp\{(e^{\lambda}-1)\phi_{2}^{*}\phi_{1}\}\langle\phi_{2}|\phi_{1}\rangle$ [28]. In the continuum limit, we take $\bar{n}_{0}\to n_{0}\Delta x^{d}$ and obtain

$$S_{i}=\ln Z(0)-\int d{\bf x}\left[\bar{\phi}({\bf x},0)e^{-\beta U({\bf x},0)}n_{0}-\bar{\phi}({\bf x},0)\phi({\bf x},0)\right]$$

(57)

To derive a simple expression for the initial action with interacting particles, it will be necessary to introduce a Hubbard-Stratonovich transformation, which we will describe below.

Finally, we note that in the field theory the work term (32) becomes

$$W[\bar{\phi},\phi]=\int_{0}^{t_{f}}dt\int d{\bf x}\,\dot{U}(t)\bar{\phi}({\bf x},t)\phi({\bf x},t).$$

(58)

The bulk action (51) can be transformed via the Cole-Hopf transformation $\phi\to e^{-\bar{\rho}}\rho$, $\bar{\phi}\to e^{\bar{\rho}}$ to

$\displaystyle S_{B}=\int_{0}^{t_{f}}dt\int d{\bf x}\biggl{[}\bar{\rho}(\partial_{t}-D\nabla^{2})\rho-\gamma\bar{\rho}\nabla\cdot(\rho\nabla\epsilon)-\gamma D\rho(\nabla\bar{\rho})^{2}\biggr{]}$

(59)

with the effective potential

$$\epsilon({\bf x})=U({\bf x})+\int d{\bf y}V({\bf x}-{\bf y})\rho({\bf y}).$$

(60)

This matches the Dean-Kawasaki field theory [20] obtained from the Langevin equation for interacting particles derived by Dean [21], then mapped to a field theory using the MSRJD technique [22, 23, 24]. However, there appear to be discrepancies with the initial and final terms. In particular, Dean-Kawasaki field theory has no analog of the projection state and final action $S_{f}$. Further the Cole-Hopf transformation does not map our initial action to the generally assumed initial conditions for Dean-Kawasaki field theory.

When $V=0$, the bulk action (51) can be written as

$$S_{B}=\int_{0}^{t_{f}}dt\int d{\bf x}\biggl{[}\bar{\phi}\partial_{t}\phi+De^{-\beta U}\nabla\bar{\phi}\cdot\nabla(\phi e^{\beta U})\biggr{]}$$

(61)

with $S_{i}$ and $S_{f}$ given by Eqs. (57) and (54). The time-reversal symmetry of the action is revealed by the transformation

$$\phi\to n_{0}e^{-\beta U}\bar{\psi}\qquad\bar{\phi}\to n_{0}^{-1}e^{\beta U}\psi\qquad t\to t^{\prime}=t_{f}-t$$

(62)

This gauge-like transformation maintains the bilinear product $\bar{\phi}\phi\to\bar{\psi}\psi$ and the action (61) is invariant apart from the time-derivative term. Suppressing the spatial integral, we have

$$\int_{0}^{t_{f}}dt\,\bar{\phi}\partial_{t}\phi\to(\bar{\psi}\psi)\bigg{|}_{t=0}^{t=t_{f}}+\int_{0}^{t_{f}}dt^{\prime}\biggl{(}\bar{\psi}\partial_{t^{\prime}}\psi+\beta\frac{\partial U}{\partial t^{\prime}}\bar{\psi}\psi\biggr{)}$$

(63)

where we have used integration by parts. Note that time reversal generates a work term (58).

The time reversal transformation (62) also maps the initial action (57) into the final action (54) and vice-versa, with the help of the boundary terms in (63). The one exception is the partition function factor, which remains $Z(0)^{-1}$ although the time-reversed averaging would require that we start at $t^{\prime}=0$ with $Z(t=t_{f})^{-1}$. Adjusting for the partition function and combining the bulk, initial, and final actions, we obtain

$$S[\bar{\phi},\phi]\to S_{R}[\bar{\psi},\psi]+\beta W_{R}+\beta F(t=t_{f})-\beta F(t=0),$$

(64)

where $S_{R}$ and $W_{R}$ are the time-reversed action and work term, and $\beta F=-\ln Z$.

As a consequence of this time reversal symmetry we have the identity

	$\displaystyle\langle Ae^{-\beta W}\rangle$	$\displaystyle=\int{\cal D}(\bar{\phi},\phi)A[\bar{\phi},\phi]e^{-\beta W[\bar{\phi},\phi]}e^{-S[\bar{\phi},\phi]}$
		$\displaystyle\to e^{-\beta\Delta F}\int{\cal D}(\bar{\psi},\psi)A_{R}[\bar{\psi},\psi]e^{-S_{R}[\bar{\psi},\psi]}$
		$\displaystyle=e^{-\beta\Delta F}\langle A_{R}\rangle_{R}$		(65)

for any observable $A[\bar{\phi},\phi]$, with time-reversed $A_{R}[\bar{\psi},\psi]$. Here we have used that work (58) is odd under time reversal, $W[\bar{\phi},\phi]\to-W_{R}[\bar{\psi},\psi]$, to move the exponential of negative work to the forward time average. We will show Eq. (65) holds for interacting particles as well, and provides the basis for deriving multiple nonequilibrium identities.

In particular, setting $A=1$ produces the Jarzynski relation, while taking $A=e^{(iq+\beta)W}$ reproduces Eq. (43), which can be inverse Fourier transformed to give the Crooks relation.

For interacting particles the bulk action (51) can be written as

$$S_{B}=\int_{0}^{t_{f}}dt\int d{\bf x}\biggl{[}\bar{\phi}\partial_{t}\phi+De^{-\beta\epsilon}\nabla\bar{\phi}\cdot\nabla(\phi e^{\beta\epsilon})\biggr{]}$$

(66)

where $\epsilon({\bf x},t)$ is the effective potential, given by

$$\epsilon({\bf x},t)=U({\bf x},t)+\int d{\bf y}\,V({\bf x}-{\bf y})\bar{\phi}({\bf y},t)\phi({\bf y},t).$$

(67)

To affect time reversal with a local transformation like (62), we need to employ a field Hubbard-Stratonovich transformation:

	$\displaystyle\exp$	$\displaystyle\biggl{(}\beta\int d{\bf y}V({\bf x}-{\bf y})\bar{\phi}({\bf y})\phi({\bf y})\biggr{)}$
		$\displaystyle=A\int{\cal D}\eta\exp\biggl{\{}{-\frac{1}{2\beta}}\int d{\bf r}_{1}d{\bf r}_{2}\,\eta^{*}({\bf r}_{1})V^{-1}({\bf r}_{12})\eta({\bf r}_{2})$
		$\displaystyle\quad+\int d{\bf r}_{1}\,\eta({\bf r}_{1})\bar{\phi}({\bf r}_{1})\phi({\bf r}_{1})\biggr{\}}e^{-i\eta_{2}({\bf x})}$
		$\displaystyle\equiv\int{\cal D}\eta F[\eta,\bar{\phi}\phi]e^{-i\eta_{2}({\bf x})}$		(68)

where $\eta({\bf x},t)=\eta_{1}({\bf x},t)+i\eta_{2}({\bf x}.t)$ is a complex field, $A$ is a normalization constant, ${\bf r}_{12}={\bf r}_{1}-{\bf r}_{2}$, and $V^{-1}$ is defined formally via $\int d{\bf y}V({\bf x}-{\bf y})V^{-1}({\bf y}-{\bf z})=\delta^{(d)}({\bf x}-{\bf z})$. Eq. (68) applies for all times $t$, with $\eta({\bf r},t)$ uncorrelated for different times. With this notation, the bulk action becomes

$$S_{B}=\int{\cal D}\eta\,F\int dtd{\bf x}\biggl{[}\bar{\phi}\partial_{t}\phi+De^{-\beta\epsilon}\nabla\bar{\phi}\cdot\nabla(\phi e^{\beta U-i\eta_{2}})\biggr{]}$$

(69)

where we have suppressed the time integration limits. Now the gauge-like field transformation

$$\phi\to n_{0}e^{-\beta U+i\eta_{2}}\bar{\psi}\qquad\bar{\phi}\to n_{0}^{-1}e^{\beta U-i\eta_{2}}\psi$$

(70)

with time reversal $t\to t^{\prime}=t_{f}-t$ once again leaves the action invariant, i.e.,

$$\nabla\bar{\phi}\cdot\nabla(\phi e^{\beta U-i\eta_{2}})\to\nabla(\psi e^{\beta U-i\eta_{2}})\cdot\nabla\bar{\psi}$$

(71)

apart from the time derivative term, which again has the form given in Eq. (63), generating the work term. Importantly, the time derivative on the $\eta$ field averages to zero because the field is uncorrelated over time.