Jarzynski Equality for an Energy-Controlled System

Abstract

The Jarzynski equality (JE) is known as an exact identity for nonequillibrium systems [1]. The JE was originally formulated for isolated and isothermal systems, while Adib reported an JE extended to an isoenergetic process [2]. On an isoenergetic process, the functional form of the Hamiltonian is altered with the energy fixed at a constant value E under the effect of an aritificial field term which is added to the Hamilton dynamics. This kind of JE is useful to calculate the entropy which replaces the Helmholtz free energy appearing in the original JE.On the other hand, the limitation of the constant energy seems to be an obstacle to widely apply the JE to more interdisciplinary problems such as optimization problems. Another problem is that it takes too long time to estimate some physical quantities since the entropy at the different energy values cannot be calculated at the same time with the JE. In the present paper, we extend the JE to an energy-controlled system in order to overcome these difficulties. In our study, we make it possible to control the instantaneous value of the energy arbitrarily in a nonequilibrium process. Under our extension, the new JE is more practical and useful to calculate the number of states and the entropy than the isoenergetic one. Furthermore, we expect that our JE can be used also for applications to computation and information science beyond physics. We will show an instance of possible application of our JE to a kind of optimization problems which is equivalent to physical problems to find the ground state [3].

Colloquium: Quantum fluctuation relations: Foundations and applications

“…A classical microcanonical version of the Jarzynski equality was put forward by Adib (2005) for non-Hamiltonian iso-energetic dynamics. It was recently generalized to energy controlled systems by Katsuda and Ohzeki (2011).…”

Erratum:Colloquium: Quantum fluctuation relations: Foundations and applications [Rev. Mod. Phys.83, 771 (2011)]

“…A classical microcanonical version of the Jarzynski equality was put forward by Adib (2005) for non-Hamiltonian isoenergetic dynamics. It was recently generalized to energy controlled systems by Katsuda and Ohzeki (2011).…”