Straightforward quantum-mechanical derivation of the Crooks fluctuation theorem and the Jarzynski equality

We obtain the Crooks and the Jarzynski nonequilibrium fluctuation relations using a direct quantum-mechanical approach for a finite system that is either isolated or coupled not too strongly to a heat bath. These results were hitherto derived mostly in the classical limit. The two main ingredients in the picture are the time-reversal symmetry and the application of the first law to the case where an agent performs work on the system. No further assumptions regarding stochastic or Markovian behavior are necessary, neither a master equation or a classical phase-space picture are required. The simplicity and the generality of these nonequilibrium relations are demonstrated, giving very simple insights into the physics.

Figure 1

The system is a gas of particles in a box. The box region is indicated by gray. A representative trajectory is illustrated. The agent on which work is being done is a piston that is free to move to infinity. After the piston is pushed out the gas particles stay in the box and no longer interact with the piston but possibly may interact (say) with a bath or with other agents. At the end of each single run of the experiment, there is an unlimited time to measure the energy of the freely moving piston in the desired resolution.

Figure 2

Illustration of a multistage process that consists of time intervals during which the system interacts with a bath and with two different agents. The system consists of gas particles that are confined to move in the shaded area. The interaction with the first agent is as described in Fig. 1. The second agent compresses that gas until the interaction with it is switched off: then it becomes like a free particle.