2020/03/03 by Ilki Kim, Kim, Ilki
Physics and Astronomy · #FOS: Physical sciences #Quantum Physics (quant-ph) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech #quant-ph
paper · pdf · doi:10.48550/arxiv.2003.01264
37 pages, 8 figures
arxiv created 2020/03/07 · arxiv updated 2020/03/10
In this work, we study the fluctuation relation and the second law of thermodynamics within a quantum linear oscillator externally driven over the period of time t = tau. To go beyond the standard approach (the two-point projective measurement one) to this subject and also render it discussed in both quantum and classical domains on the single footing, we recast this standard approach in terms of the Wigner function and its propagator in the phase space (x,p). With the help of the canonical transformation from (x,p) to the angle-action coordinates (ϕ,I), we can then derive a measurement-free (classical-like) form of the Crooks fluctuation relation in the Wigner representation. This enables us to introduce the work W_I0,Itau associated with a single run from (I0) to (Itau) over the period tau, which is a quantum generalization of the thermodynamic work with its roots in the classical thermodynamics. This quantum work differs from the energy difference e_I0,Itau = e(Itau) - e(I0) unless beta, hbar --> 0. Consequently, we will obtain the quantum second-law inequality Delta Fbeta ≤ <W>P ≤ <e>P = Delta U, where P, Delta Fbeta, and <W>P denote the work (quasi)-probability distribution, the free energy difference, and the average work distinguished from the internal energy difference Delta U, respectively, while <W>P --> Delta U in the limit of beta, hbar --> 0 only. Therefore, we can also introduce the quantum heat Qq = Delta U - W even for a thermally isolated system, resulting from the quantum fluctuation therein. This is a more fine-grained result than <W>P = Delta U obtained from the standard approach. Owing to the measurement-free nature of the thermodynamic work W_I0,Itau, our result can also apply to the (non-thermal) initial states rho0 = (1-gamma) rhobeta + gamma sigma with sigma ≠ rhobeta.