Approach to Equilibrium in Nano-scale Systems at Finite Temperature
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概要
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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.
- 2010-12-15
著者
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De Raedt
Department Of Applied Physics Materials Science Center University Of Groningen
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Miyashita Seiji
Department Of Applied Physics Graduate School Of Engineering The University Of Tokyo
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Yuan Shengjun
Institute of Molecules and Materials, Radboud University of Nijmegen, NL-6525ED Nijmegen, The Netherlands
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Katsnelson Mikhail
Institute of Molecules and Materials, Radboud University of Nijmegen, NL-6525ED Nijmegen, The Netherlands
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Michielsen Kristel
Institute for Advanced Simulation, Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52425 Jülich, Germany
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Michielsen Kristel
Institute for Advanced Simulation, Jülich Supercomputing Centre, Research Centre Jülich, D-52425 Jülich, Germany
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Jin Fengping
Department of Applied Physics, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747AG Groningen, The Netherlands
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