Electromagnetic gauge-freedom and work

Abstract

We argue that the definition of the thermodynamic work done on a charged particle by a timedependent electromagnetic field is an open problem, because the particle's Hamiltonian is not gaugeinvariant. The solution of this problem demands accounting for the source of the field. Hence we focus on the work done by a heavy body (source) on a lighter particle when the interaction between them is electromagnetic and relativistic. The work can be defined via the gauge-invariant kinetic energy of the source. We uncover a formulation of the first law (or the generalized work-energy theorem) which is derived from relativistic dynamics, has definite validity conditions, and relates the work to the particle's Hamiltonian in the Lorenz gauge. Thereby the thermodynamic work also relates to the mechanic work done by the Lorentz force acting on the source. The formulation of the first law is based on a specific separation of the overall energy into those of the source, particle and electromagnetic field. This separation is deduced from a consistent energy-momentum tensor. Hence it holds relativistic covariance and causality.

Free energy amplification by magnetic flux for driven quantum systems

“…In addition, we have not considered the time-dependent magnetic field 62 , but the present free energy amplification can appear even for a constant magnetic field. Under a time-dependent electromagnetic field, the definition of the thermodynamic work on a charged particle needs a special care 63 . The generalization of the present free energy equal to the case with the time-dependent magnetic field is an intriguing direction.…”

Jarzynski Equality for Driven Quantum Field Theories

“…To date another decade has gone by, yet quantum stochastic thermodynamics is still rather incomplete. How to describe thermodynamic work and entropy production in open quantum systems is still hotly debated [46][47][48][49][50][51][52][53][54][55][56], and with a few exceptions [57][58][59][60][61] most of the literature is restricted to standard Schrödinger quantum mechanics. The purpose of the present analysis is to significantly broaden the scope of stochastic thermodynamics, and to take the next, important step -extend quantum stochastic thermodynamics to quantum field theories.…”

Defining the Work Done on an Electromagnetic Field

“…Note that we always deal with the full energy (space-integrated) energy of EMF. The localization of this energy is not studied; this is another (and more difficult) problem [24].…”

“…Using (24) we introduce the canonical momentum Π k (x) = −∆ −1 ( Ḃk )(x), and construct from (21) the Hamiltonian via the usual Legendre transformation…”