Nonequilibrium work distribution

Work as a trajectory functional

Consider a system with microstate $x$ (classical phase point or configuration) and a parameter-dependent Hamiltonian/energy $H(x,\lambda)$. A driving protocol $\lambda_t$ is applied over $t\in[0,\tau]$.

For a single realization with trajectory $t\mapsto x_t$, the (protocol) work done on the system is commonly defined as

For discrete-time driving $\lambda_{0},\lambda_{1},\dots,\lambda_{N}$ with microstates $x_0,\dots,x_N$,

i.e. the energy change due purely to parameter updates.

Along a single realization, one can decompose the energy change as

where $\Delta E$ is the change in internal energy (compare internal energy ) and $Q$ is heat exchanged with the environment (first-law bookkeeping).

The work distribution

Assume an initial distribution for $x_0$ (often equilibrium, e.g. the canonical ensemble at $\lambda_0$) and a dynamics generating a path measure on trajectories (Hamiltonian dynamics, Langevin, a Markov chain , or a Markov semigroup ).

The work distribution $P(W)$ is the pushforward of the path measure under the map $\omega\mapsto W(\omega)$. Formally,

where $\mathbb{P}$ is the path probability measure and $\delta$ is the Dirac delta.

The mean work is $\langle W\rangle$, an expectation under $P(W)$.

Forward and reverse distributions

For a protocol $\lambda_t$ (forward) and its time reversal $\tilde\lambda_t=\lambda_{\tau-t}$ (reverse), denote the distributions by $P_F(W)$ and $P_R(W)$. Their relation is the content of the Crooks fluctuation theorem and implies Jarzynski’s equality .

Large-deviation viewpoint (optional)

In repeated driving of large systems, $W$ (or work per particle) often satisfies a large deviation principle with a rate function , and fluctuation relations constrain that rate function via symmetry identities.