Time Scales in the Approach to Equilibrium of Macroscopic Quantum Systems

Abstract

We prove two theorems concerning the time evolution in general isolated quantum systems. The theorems are relevant to the issue of the time scale in the approach to equilibrium. The first theorem shows that there can be pathological situations in which the relaxation takes an extraordinarily long time, while the second theorem shows that one can always choose an equilibrium subspace the relaxation to which requires only a short time for any initial state. The recent renewed interest in the foundation of quantum statistical mechanics and in the dynamics of isolated quantum systems has led to a revival of the old approach by von Neumann to investigate the problem of thermalization only in terms of quantum dynamics in an isolated system [1,2]. It has been demonstrated in some general or concrete settings that a pure initial state evolving under quantum dynamics indeed approaches an equilibrium state [3][4][5][6][7][8][9][10]. The underlying idea that a single pure quantum state can fully describe thermal equilibrium has also become much more concrete [11][12][13].We must note, however, that in the general theories of the approach to equilibrium [1][2][3][4][5][6][7][8][9], the issue of the time scale required for the relaxation has not been fully addressed. 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.In the present paper we prove two theorems for a general class of isolated macroscopic quantum systems. Although the theorems may look somewhat artificial, they are of direct physical relevance to the above mentioned issue of the time scale as we shall explain below.Our first theorem is a warning; it states that there always exists an equilibrium subspace for which the relaxation to equilibrium takes an extraordinarily long time. Although this is nothing more than a purely theoretical "existence proof", it shows that the general theories [1][2][3][4][5][6][7][8][9] should be supplemented by extra arguments that guarantee the necessary time scale to be sufficiently short.Our second theorem, which has the opposite character, gives us a hope; it states that there can be an equilibrium subspace for which any initial state approaches equilibrium within a short amount of time. Although the subspace we shall construct is artificial, it is expected that a realistic equilibrium subspace shares some essential features with our example.We hope that the present results serve as a basis of future investigation of the fundamental and important problem of the approach to equilibrium [16].Setup and background .-We consider an abstract model for an isolated macroscopic quantum system in a large finite volume V . A typ...

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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]…”

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