Abstract
Systems that are driven out of thermal equilibrium typically dissipate random quantities of energy on microscopic scales. Crooks fluctuation theorem relates the distribution of these random work costs with the corresponding distribution for the reverse process. By an analysis that explicitly incorporates the energy reservoir that donates the energy, and the control system that implements the dynamic, we here obtain a quantum generalization of Crooks theorem that not only includes the energy changes in the reservoir, but the full description of its evolution, including coherences. This approach moreover opens up for generalizations of the concept of fluctuation relations. Here we introduce `conditional' fluctuation relations that are applicable to non-equilibrium systems, as well as approximate fluctuation relations that allow for the analysis of autonomous evolution generated by global time-independent Hamiltonians. We furthermore extend these notions to Markovian master equations, implicitly modeling the influence of the heat bath.
Highlights
Imagine a physical system with a Hamiltonian HS ðxÞ that depends on some external parameter x, e.g., electric or magnetic fields that we can vary at will
There are several reasons why it is useful to generalize the type of quantum fluctuation theorems that we have considered so far
This is in contrast to approaches where the purpose of the measurements is to generate classical outcomes and where the quantum fluctuation relations are based on the resulting probability distributions, much in the spirit of classical fluctuation relations
Summary
Imagine a physical system with a Hamiltonian HS ðxÞ that depends on some external parameter x, e.g., electric or magnetic fields that we can vary at will. We can push the system out of thermal equilibrium This would typically require work that may be dissipated because of interactions with the surrounding heat bath. The key is to explicitly model all degrees of freedom (d.o.f.) involved in the process This includes the control mechanism that implements the change of the external parameter, i.e., x in HS ðxÞ, as well as the “energy reservoir,”. The general theme of this investigation is to formulate fluctuation relations in terms of this induced evolution. To make this more concrete, let us briefly display the first of the fluctuation theorems that we will derive.
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