Жарыңски теңдеуі: Статистикалық механикадағы еріксіз жұмыс және еркін энергия айырмашылығы

Equation in statistical mechanics
The Jarzynski equality (JE) is an equation in statistical mechanics that relates free energy differences between two states and the irreversible work along an ensemble of trajectories joining the same states. It is named after the physicist Christopher Jarzynski (then at the University of Washington and Los Alamos National Laboratory, currently at the University of Maryland) who derived it in 1996. Fundamentally, the Jarzynski equality points to the fact that the fluctuations in the work satisfy certain constraints separately from the average value of the work that occurs in some process.

Taking the log of , and use the cumulant expansion up to the second cumulant, we obtain The left side is the work dissipated into the heat bath, and the right side could be interpreted as the fluctuation in the work due to thermal noise. Consider dragging an overdamped particle in a viscous fluid with temperature at constant force for a time Because there is no potential energy for the particle, the change in free energy is zero, so we obtain
The work expended is , where is the total displacement during the time. The particle's displacement has a mean part due to the external dragging, and a varying part due to its own diffusion, so , where is the diffusion coefficient. Together, we obtainor , where is the viscosity. This is the fluctuation dissipation theorem. In fact, for most trajectories, the work is positive, but for some rare trajectories, the work is negative, and those contribute enormously to the expectation, giving us an expectation that is exactly one.

A question has been raised about who gave the earliest statement of the Jarzynski equality. For example, in 1977 the Russian physicists G. N. Bochkov and Yu. E. Kuzovlev (see Bibliography) proposed a generalized version of the fluctuation dissipation theorem which holds in the presence of arbitrary external time dependent forces. Despite its close similarity to the JE, the Bochkov Kuzovlev result does not relate free energy differences to work measurements, as discussed by Jarzynski himself in 2007. Another similar statement to the Jarzynski equality is the nonequilibrium partition identity, which can be traced back to Yamada and Kawasaki. (The Nonequilibrium Partition Identity is the Jarzynski equality applied to two systems whose free energy difference is zero like straining a fluid.) However, these early statements are very limited in their application. Both Bochkov and Kuzovlev as well as Yamada and Kawasaki consider a deterministic time reversible Hamiltonian system. As Kawasaki himself noted this precludes any treatment of nonequilibrium steady states. The fact that these nonequilibrium systems heat up forever because of the lack of any thermostatting mechanism leads to divergent integrals etc. No purely Hamiltonian description is capable of treating the experiments carried out to verify the Crooks fluctuation theorem, Jarzynski equality and the fluctuation theorem. These experiments involve thermostatted systems in contact with heat baths.