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arXiv:0904.1501 · Journal of the Physical Society of Japan

Origin of the Canonical Ensemble: Thermalization with Decoherence

Shengjun Yuan, Mikhail I. Katsnelson, Hans A. De Raedt

Published Aug 28, 2009• 32 citations
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Abstract

We solve the time-dependent Schrodinger equation for the combination of a spin system interacting with a spin bath environment. In particular, we focus on the time development of the reduced density matrix of the spin system. Under normal circumstances we show that the environment drives the reduced density matrix to a fully decoherent state, and furthermore the diagonal elements of the reduced density matrix approach those expected for the system in the canonical ensemble. We show one exception to the normal case is if the spin system cannot exchange energy with the spin bath. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the canonical ensemble is a state that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics.

DOI
10.1143/jpsj.78.094003

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•Jan 1, 2008•0 citations
We solve the time-dependent Schrodinger equation for the combination of a spin system interacting with a spin bath environment. In particular, we focus on the time development of the reduced density matrix of the spin system. Under normal circumstances we show that the environment drives the reduced density matrix to a fully decoherent state, and furthermore the diagonal elements of the reduced density matrix approach those expected for the system in the canonical ensemble. We show one exception to the normal case is if the spin system cannot exchange energy with the spin bath. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the canonical ensemble is a state that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics.
In this review, we discuss the decoherence and thermalization of a quantum spin system interacting with a spin bath environment, by numerically solving the time-dependent Schrödinger equation of the whole system. The effects of the topologic structure and the initial state of the environment on the decoherence of the two-spin and many-spin system are discussed. The role of different spin-spin coupling is considered. We show under which conditions the environment drives the reduced density matrix of the system to a fully decoherent state, and how the diagonal elements of the reduced density matrix approach those expected for the system in the microcanonical or canonical ensemble, depending on the character of the additional integrals of motion. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the microcanonical distribution (in each eigenenergy subspace) and canonical ensemble are states that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics. Furthermore, our numerical results show that a fully decoherent quantum system prefers to stay in an equilibrium state with a maximum entropy, indicating the validity of the second law of thermodynamics in the decoherence process.
We consider a system weakly interacting with a bath as a thermodynamic setting to establish a quantum foundation of statistical physics. It is shown that even if the composite system is initially in an arbitrary nonequilibrium pure quantum state, the unitary dynamics of a generic weak interaction almost always drives the subsystem into the canonical ensemble, in the usual sense of typicality. A crucial step is taken by assuming that the matrix elements of the interaction Hamiltonian have random phases, while their amplitudes are left unrestricted.
We present an exactly solvable model to study the role of the system-bath coupling for the generalized canonical thermalization, which reduces almost all the pure states of the ``universe'' (formed by a system SSS plus its surrounding heat bath BBB) to a canonical equilibrium state of SSS. It is found that, for the overwhelming majority of the ``universe'' states (they are entangled at least), the diagonal canonical typicality remains robust with respect to finite interactions between SSS and BBB. Particularly, a decoherence mechanism is utilized here to account for the vanishing of the off-diagonal elements of the reduced density matrix of SSS. The nonvanishing off-diagonal elements due to the finite size of the bath and the stronger system-bath interaction might offer more to quantum thermalization.
The role of the system-bath coupling is studied for the generalized canonical thermalization of a system S surrounded by a heat bath B, which was most recently shown [S. Popescu, et al., Nature Physics 2,754(2006) and S. Goldstein et al., Phys. Rev. Lett. 96, 050403(2006)] as a kinematic reduction to a canonical equilibrium state from almost all the pure states of the universe formed by S plus B. This kinematic process is revisited by explicitly considering the energy shell deformation due to the interaction between S and B with an exactly solvable model. By calculating the state numbers of the universe and its subsystems S and B in the deformed energy shells, it is found that, for the overwhelming majority of the universe states (they are entangled at least), the diagonal canonical typicality remains robust with respect to finite interactions between S and B. Particularly, the kinematic decoherence is utilized here to account for the vanishing of the off-diagonal elements of the reduced density matrix of S. It is pointed out that the non-vanishing off-diagonal elements due to the finiteness of bath and the stronger system-bath interaction might offer novelties of the quantum thermalization.
We study the time evolution of the reduced density matrix of a system of spin-1/2 particles interacting with an environment of spin-1/2 particles.The initial state of the composite system is taken to be a product state of a pure state of the system and a pure state of the environment.The latter pure state is prepared such that it represents the environment at a given finite temperature in the canonical ensemble.The state of the composite system evolves according to the time-dependent Schrödinger equation, the interaction creating entanglement between the system and the environment.It is shown that independent of the strength of the interaction and the initial temperature of the environment, all the eigenvalues of the reduced density matrix converge to their stationary values, implying that also the entropy of the system relaxes to a stationary value.We demonstrate that the difference between the canonical density matrix and the reduced density matrix in the stationary state increases as the initial temperature of the environment decreases.As our numerical simulations are necessarily restricted to a modest number of spin-1/2 particles (< 36), but do not rely on time-averaging of observables nor on the assumption that the coupling between system and environment is weak, they suggest that the stationary state of the system directly follows from the time evolution of a pure state of the composite system, even if the size of the latter cannot be regarded as being close to the thermodynamic limit.
We analyze the dynamics of N interacting spins (quantum register) collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part. We find the decay rates of the reduced density matrix elements in the energy basis. We show that if the spins do not interact among each other, then the fastest decay rates of off-diagonal matrix elements induced by the energy conserving inter-action is of order N2, while that one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. For a general spin system the decay rates depend in a rather complicated (but explicit) way on the size N and the interaction between the spins. Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings. We do not make Markov-, Born- or weak coupling (van Hove) approximations. 1