Thermodynamics under Dynamic Forcing

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Abstract
Beginning with a system that is governed by an arbitrary time‐dependent Hamiltonian, we exhibit an existence proof for a unitary generator that has an arbitrary initial value and yet contact transforms the representation to one governed by any given kinematically equivalent Hamiltonian. By choosing the initial value of the unitary operator to be unity, we are able to compare the behaviour of the same system under two different Hamiltonians and the same initial state vector. We thus are able to establish that the eventual physical states evolving from two distinct initial quantum state vectors will become practically indistinguishable under one of the two Hamiltonians if and only if they do so under the other. For the restricted class of systems for which one of the two Hamiltonians is a time‐independent energy operator, and also generates equilibrium thermodynamics, then the condition for merging under the time‐dependent Hamiltonian is the same as under the time‐independent one. The two states must have the same initial energy. As a special case of the above, we choose the time‐independent Hamiltonian to be the relativistic energy measuring operator for the time‐dependent Hamiltonian, as associated with the chosen initial time. If the system under the time‐dependent Hamiltonian is such that its relativistic energy measuring operator for any fixed time generates equilibrium thermodynamics, then we are led rigorously to the conclusion that the instantaneous relativistic energy for the system under the time‐dependent Hamiltonian is simply a well‐defined function of time and depends only on the initial energy and not on any other initial conditions. For a composite system that is of the above type, and in addition consists of one very small system in contact with a very large one, which is called a generalized reservoir, we consider a specific initial physical state for the large system, and various states for the small one. The eventual dynamic state of the composite system is essentially independent of the initial state of the small system which has almost no influence on the total composite energy. Hence the eventual dynamic state of the small system is shown rigorously to be independent of its initial state. For a forced system with a time‐dependent Hamiltonian, we discuss the assignment of equilibrium thermodynamic potentials to a representation with a time‐independent Hamiltonian. We discuss the concept of a process under a time‐dependent Hamiltonian. Such a process is a natural generalization of the static and quasi‐static processes. Also, we verify all of the theory with both general and specific examples of electromagnetic interactions.