Dissipation: The Phase-Space Perspective

We show, through a refinement of the work theorem, that the average dissipation, upon perturbing a Hamiltonian system arbitrarily far out of equilibrium in a transition between two canonical equilibrium states, is exactly given by $⟨W_{\mathrm{diss}}⟩=⟨W⟩-\Delta F=kTD(\rho \parallel \tilde{\rho })=kT⟨\ln (\rho /\tilde{\rho })⟩$, where $\rho$ and $\tilde{\rho }$ are the phase-space density of the system measured at the same intermediate but otherwise arbitrary point in time, for the forward and backward process. $D(\rho \parallel \tilde{\rho })$ is the relative entropy of $\rho$ versus $\tilde{\rho }$. This result also implies general inequalities, which are significantly more accurate than the second law and include, as a special case, the celebrated Landauer principle on the dissipation involved in irreversible computations.

Figure 1

Space-time coordinates for the forward and backward trajectories.

Figure 2

Dissipation of Brownian particles in a harmonic potential with spring constant varying from $\kappa =2$ to ${\kappa }^{\prime }=1$ during a time interval $\tau$. The solid line is dissipative work directly obtained from simulation. Dashed and dotted lines indicate the lower bound of the dissipative work estimated from the relative entropy.

Figure 3

Top: Cyclic variation of a model potential. The arrows indicate the forward protocol corresponding to the Szilard engine. The backward process is a model for recording and erasing information. Bottom: Partitioned average work as a function of processing time $\tau$: $⟨W{⟩}_R$ (solid line) and $-⟨\tilde{W}{⟩}_R$ (dashed line). The bound (8) using the coarse-grained distribution is shown by the dotted line.