門内 隆明

<p>近年，非平衡過程を記述する「ゆらぎ定理」が注目を集めている．この定理は通常，熱平衡状態にある系にする仕事に注目したものであるが，本研究では，非平衡定常状態にある量子系に操作をしてゆらぎ定理が成立しているかを議論する．この際，可解モデルを採用し厳密に解析する．その結果，熱平衡状態ではゆらぎ定理が成立していたが，非平衡定常状態では成立しないことがわかり，非平衡定常状態でも成立するような物理量を探った．</p>

We show that it is possible to calculate equilibrium expectation values of many-body correlated quantities such as the characteristic functions and probability distributions with the use of only a single typical pure state. It also means that we can apply the pure state approach to Heisenberg operators and their spectral fluctuation, and hence to nonequilibrium processes starting from equilibrium. In particular, we can accurately analyze the full statistics of entropy production in nonequilibrium mesoscopic systems. In this way, we can access the full information on higher-order fluctuations in the large deviation regime far from equilibrium.

We derive analytical formulas for the firing rate of integrate-and-fire neurons endowed with realistic synaptic dynamics. In particular, we include the possibility of multiple synaptic inputs as well as the effect of an absolute refractory period into the description. The latter affects the firing rate through its interaction with the synaptic dynamics.

Large fluctuation of Brownian particles is affected by the finiteness of the correlation length of the background noise field. Indeed a Fokker-Planck equation is derived in a Markovian limit of a spatio-temporal short correlated noise. Corresponding kinetic quantities are renormalized due to the spatio-temporal memory. We also investigate the case of open system by Connecting a thermostat to the system.

Large fluctuation of Brownian particles is affected by the finiteness of the correlation length of the background noise field. Indeed a Fokker-Planck equation is derived in a Markovian limit of a spatio-temporal short correlated noise. Corresponding kinetic quantities are renormalized due to the spatio-temporal memory. We also investigate the case of open system by connecting a thermostat to the system. © 2009 Springer Science+Business Media B.V.

Spatio-temporal generalization of the Ornstein-Uhlenbeck process has been received considerable attention in the context of coagulations in a random flow. We shall explore the diffusion process with a symmetric spatio-temporal correlated noise in the presence of the external force for the underdamped case. In a nontrivial short correlation limit, the role of spatial correlation is explored and a Fokker-Planck equation is derived based on the stochastic Liouville equation. Our Fokker-Planck equation interpolates the recently proposed diffusion equation describing the generalized Ornstein-Uhlenbeck processes and the traditional Kramers equation. In the small limit of the characteristic value of the momentum given by the mass, spatial and temporal correlation lengths, the diffusion coefficient is proportional to the inverse of the momentum. On the other hand, for the large limit of the characteristic momentum constant, the usual Brownian motion without spatial randomness is reproduced. The analytic form of the steady state distribution is numerically verified with use of the stochastic simulation. © 2009 IOP Publishing Ltd.

We explore the diffusion process in the non-Markovian spatio-temporal noise. There is a non-trivial short-memory regime, i.e., the Markovian limit characterized by a scaling relation between the spatial and temporal correlation lengths. In this regime, a Fokker-Planck equation is derived by expanding the trajectory around the systematic motion and the non-Markovian nature amounts to the systematic reduction of the potential. For a system with the potential barrier, this fact leads to the renormalization of both the barrier height and collisional prefactor in the Kramers escape rate, with the resultant rate showing a maximum at some scaling limit. Copyright (C) EPLA, 2008

We investigate the low-temperature relaxation dynamics toward a nonequilibrium steady state in a tilted asymmetric periodic potential based on the WKB analysis and the numerical diagonalization of the Fokker-Planck operator. Due to the tilting, the Fokker-Planck operator, and thus the Schrodinger operator associated with it, are non-Hermitian. Therefore, we evaluate the decay rate based on the WKB analysis both for real- and complex-valued eigenvalues. In the tilting range where the double-humped barrier exists, the decay rate is shown to obey a law which is a subtle nonequilibrium extension of the so-called Kramers escape rate. The decay rate for the single-humped barrier case is analyzed as well. The large tilting regime where the barriers no longer exist is also investigated.

We investigate the low-temperature relaxation dynamics toward a nonequilibrium steady state in a tilted asymmetric periodic potential based on the WKB analysis and the numerical diagonalization of the Fokker-Planck operator. Due to the tilting, the Fokker-Planck operator, and thus the Schrodinger operator associated with it, are non-Hermitian. Therefore, we evaluate the decay rate based on the WKB analysis both for real- and complex-valued eigenvalues. In the tilting range where the double-humped barrier exists, the decay rate is shown to obey a law which is a subtle nonequilibrium extension of the so-called Kramers escape rate. The decay rate for the single-humped barrier case is analyzed as well. The large tilting regime where the barriers no longer exist is also investigated.

We investigate asymptotic decay phenomena towards the nonequilibrium steady state of the thermal diffusion in a periodic potential in the presence of a constant external force. The parameter dependence of the decay rate is revealed by investigating the Fokker-Planck (FP) equation in the low temperature case under the spatially periodic boundary condition (PBC). We apply the WKB method to the associated Schrodinger equation. While eigenvalues of the non-Hermitian FP operator are complex in general, in a small tilting case accompanied with local minima, the imaginary parts of the eigenvalues are almost vanishing. Then the Schrodinger equation is solved with PBC. The decay rate is analyzed in the context of quantum tunneling through a triple-well effective periodic potential. In a large tilting case, the imaginary parts of the eigenvalues of the FP operator are crucial. We apply the complex-valued WKB method to the Schrodinger equation with the absorbing boundary condition, finding that the decay rate saturates and depends only on the temperature, the period of the potential and the damping coefficient. The intermediate tilting case is also explored. The analytic results well agree with the numerical data for a wide range of tilting. Finally, in the case that the potential includes a higher Fourier component, we report the slow relaxation, which is taken as the resonance tunneling. In this case, we analytically obtain the Kramers type decay rate.

We investigate asymptotic decay phenomena towards the nonequilibrium steady state of the thermal diffusion in a periodic potential in the presence of a constant external force. The parameter dependence of the decay rate is revealed by investigating the Fokker-Planck (FP) equation in the low temperature case under the spatially periodic boundary condition (PBC). We apply the WKB method to the associated Schrodinger equation. While eigenvalues of the non-Hermitian FP operator are complex in general, in a small tilting case accompanied with local minima, the imaginary parts of the eigenvalues are almost vanishing. Then the Schrodinger equation is solved with PBC. The decay rate is analyzed in the context of quantum tunneling through a triple-well effective periodic potential. In a large tilting case, the imaginary parts of the eigenvalues of the FP operator are crucial. We apply the complex-valued WKB method to the Schrodinger equation with the absorbing boundary condition, finding that the decay rate saturates and depends only on the temperature, the period of the potential and the damping coefficient. The intermediate tilting case is also explored. The analytic results well agree with the numerical data for a wide range of tilting. Finally, in the case that the potential includes a higher Fourier component, we report the slow relaxation, which is taken as the resonance tunneling. In this case, we analytically obtain the Kramers type decay rate.

We explore the role of an intermediate state (phase) in homogeneous nucleation by examining the decay process through a double-humped potential barrier. We analyze the one-dimensional Fokker-Planck (FP) equations with the fourth- and sixth-order Landau potentials. In the low-temperature case, we apply the WKB method to the FP equation and obtain an analytic expression for the decay rate which is accurate for a wide range of depth and curvature of the middle well. In the case of a deep middle well, it reduces to an extended Kramers formula, in which the barrier height in the original formula is replaced by the arithmetic mean height of the higher (outer) and lower (inner) barriers, and the curvature of the initial well in the original one is replaced by the geometric mean curvature of the initial and intermediate wells. In the case of a shallow middle well, the Kramers escape rate is evaluated also within the standard framework of the mean-first-passage-time problem, whose result is consistent with our WKB analysis. Criteria for enhancement of the decay rate are revealed.

We investigate the phenomenon of asymptotic decay towards the nonequilibrium steady state of the thermal diffusion in the presence of a tilted periodic potential. The parameter dependence of the decay rate is revealed by investigating the Fokker-Planck (FP) equation in the low temperature case under the spatially periodic boundary condition (PBC). We apply the WKB method to the associated Schrodinger equation. While eigenvalues of the non-Hermitian FP operator are complex in general, in a small tilting case the imaginary parts of the eigenvalues are almost vanishing. Then the Schrodinger equation is solved with PBC. The decay rate is analyzed in the context of quantum tunneling through a triple-well effective periodic potential. In a large tilting case, the imaginary parts of the eigenvalues of the FP operator are crucial. We apply the complex-valued WKB method to the Schrodinger equation with the absorbing boundary condition, finding that the decay rate saturates and depends only on the temperature, the potential periodicity and the viscous constant. The intermediate tilting case is also explored. The analytic results agree well with the numerical data for a wide range of tilting. (c) 2006 Elsevier B.V All rights reserved.

There are two related theorems which hold even in far from equilibrium, namely fluctuation theorem and Jarzynski equality. Fluctuation theorem states the existence of symmetry of fluctuation of entropy production, while the Jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes. On the other hand, the relationship between these theorems was investigated by Crooks [Phys. Rev. E 60, 2721 (1999)] for the classical stochastic systems. In this paper, we derive quantum analogues of fluctuation theorem and Jarzynski equality in terms of microscopic reversibility. In other words, the quantum analog of the work by Crooks is presented. Also, for the quasiclassical Langevin system, microscopically reversible condition is confirmed. © 2005 The American Physical Society.

非平衡系の正負エントロピー生成の生起確率の比が物質定数を含まない形でバランスすることが古典系で広く成立することが知られており、Fluctuation Theorem(FT)と呼ばれている。以前、量子系への拡張を仕事演算子を定義して行いプランク定数の2乗に比例する補正があることを特定のモデルで示した。一方、Kurchanによってこれと古典極限では一致する等価でない定式化がなされていたが両者の関係ははっきりしていなかった。今回は両者の関係を明確にする。これによって量子補正の起源が理解される。