Quantum equilibration in finite time

Abstract

It has recently been shown that small quantum subsystems generically equilibrate, in the sense that they spend most of the time close to a fixed equilibrium state. This relies on just two assumptions: that the state is spread over many different energies, and that the Hamiltonian has nondegenerate energy gaps. Given the same assumptions, it has also been shown that closed systems equilibrate with respect to realistic measurements. We extend these results in two important ways. Firstly, we prove equilibration over a finite (rather than infinite) time-interval, allowing us to bound the equilibration time. Secondly, we weaken the non-degenerate energy gaps condition, showing that equilibration occurs provided that no energy gap is hugely degenerate. 9 Acknowledgments 10 Appendix A. Proof of equation (13) 10 Appendix B. Proof of theorem 2 11 Appendix C. Proof of theorem 3 111 For a precise definition of these criteria, see the next section. 2 If desired, we could define d E to be the number of distinct energies on which ρ(t) has support. This gives a slight improvement in the results, but has the disadvantage of making d E and N (ε) depend on the state, rather than the Hamiltonian alone.

Effects of the interplay between initial state and Hamiltonian on the thermalization of isolated quantum many-body systems

“…We can then talk about equilibration in a probabilistic sense. The rate of decay of these fluctuations with system size depends on the system investigated [36][37][38][39][40][41][42][43]. The important fact for us here is that for the models we study the temporal fluctuations are indeed small and should vanish for very large systems [36].…”

Time Scales in the Approach to Equilibrium of Macroscopic Quantum Systems

“…Usually a statement about the relaxation is proved for "sufficiently long (but finite) time", but no concrete estimates are made of how long the "finite time" will be. Although there is an interesting attempt [14] to deal with the time-scale, we find their main result not very useful for large systems [15]. If it happens that the required time is as long as, say, the age of the universe, the statement about the approach to equilibrium may not be physically relevant.…”

Fluctuation Theorem for Many-Body Pure Quantum States

“…Modern researches has revealed that even a pure quantum state, described by a single wave function without any genuine thermal fluctuation, can relax to macroscopic thermal equilibrium by a reversible unitary evolution [1][2][3][4][5][6][7][8][9][10][11]. Thermalization of isolated quantum systems, which is relevant to the zeroth law of thermodynamics, is now a very active area of researches in theory [1][2][3][4][5][6], numerics [10][11][12][13][14][15][16], and experiments [17][18][19][20][21][22][23]. Especially, the concepts of typicality [9,[24][25][26] and the eigenstate thermalization hypothesis (ETH) [10][11][12][27][28][29][30][31][32][33][34][35]…”