We analyze the structure of stochastic dynamics near either a stable or unstable fixed point, where the force can be approximated by linearization. We find that a cost function that determines a Boltzmann-like stationary distribution can always be defined near it. Such a stationary distribution does not need to satisfy the usual detailed balance condition but might have instead a divergence-free probability current. In the linear case, the force can be split into two parts, one of which gives detailed balance with the diffusive motion, whereas the other induces cyclic motion on surfaces of constant cost function. By using the Jordan transformation for the force matrix, we find an explicit construction of the cost function. We discuss singularities of the transformation and their consequences for the stationary distribution. This Boltzmann-like distribution may be not unique, and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point.Boltzmann distribution ͉ cost function ͉ detailed balance ͉ cyclic motion I n equilibrium statistical mechanics, the principle of detailed balance and the related fluctuation-dissipation theorem play important roles. Einstein used the principle that the excess energy that is put into each mode of an equilibrium system in the course of thermal fluctuations is also removed from the same mode by dissipative forces. This principle is implicit in his work on Brownian movement (1), and explicit in later works on the photoelectric effect (2), and on the relation between spontaneous and induced emission of electromagnetic radiation (3). It was formulated as the principle of detailed balance by Bridgman (4) and used to explain Johnson noise in electrical circuits by Nyquist (5). It is related to the fact that the same processes that drive fluctuations in the neighborhood of a typical equilibrium configuration also drive the configuration back towards a typical equilibrium or steady-state configuration when it is displaced from equilibrium by an amount that is small, but large compared with the fluctuations in thermal equilibrium. In this situation, the equilibrium distribution in phase space is just the Boltzmann distribution, proportional to e Ϫ␤E , where ␤ is inversely proportional to temperature, and E is the energy of the point in phase space. Configurations that differ significantly from those that contribute to the minimum of the free energy are driven back to the neighborhood of this minimum by dissipative effects such as thermal or electrical conduction or viscosity, and the magnitude of these effects is related to the equilibrium fluctuations of related variables.In many situations, there is no thermodynamic equilibrium, but external steady and fluctuating forces drive the system into a steady or very slowly varying state for which the principle of detailed balance does not hold. A light bulb powered by an external battery or a chemical reaction in which the reactants are introduced at a steady ra...

We present the theoretical study on non-equilibrium (NEQ) fluctuations for diffusion dynamics in high dimensions driven by a linear drift force. We consider a general situation in which NEQ is caused by two conditions: (i) drift force not derivable from a potential function and (ii) diffusion matrix not proportional to the unit matrix, implying non-identical and correlated multi-dimensional noise. The former is a well-known NEQ source and the latter can be realized in the presence of multiple heat reservoirs or multiple noise sources. We develop a statistical mechanical theory based on generalized thermodynamic quantities such as energy, work, and heat. The NEQ fluctuation theorems are reproduced successfully. We also find the time-dependent probability distribution function exactly as well as the NEQ work production distribution P (W) in terms of solutions of nonlinear differential equations. In addition, we compute low-order cumulants of the NEQ work production explicitly. In two dimensions, we carry out numerical simulations to check out our analytic results and also to get P (W). We find an interesting dynamic phase transition in the exponential tail shape of P (W), associated with a singularity found in solutions of the nonlinear differential equation. Finally, we discuss possible realizations in experiments.

A recently developed treatment of stochastic processes leads to the construction of a potential landscape for the dynamical evolution of complex systems. Since the existence of a potential function in generic settings has been frequently questioned in literature, here we study several related theoretical issues that lie at core of the construction. We show that the novel treatment, via a transformation, is closely related to the symplectic structure that is central in many branches of theoretical physics. Using this insight, we demonstrate an invariant under the transformation.We further explicitly demonstrate, in one-dimensional case, the contradistinction among the new treatment to those of Ito and Stratonovich, as well as others. Our results strongly suggest that the method from statistical physics can be useful in studying stochastic, complex systems in general.To cite published version: P. Ao, C. Kwon, H. Qian, 2007, On the existence of potential landscape in the evolution of complex systems, Complexity 12: 19-27.

We show that the total entropy production in stochastic processes with odd-parity variables (under time reversal) is separated into three parts, only two of which satisfy the integral fluctuation theorems in general. One is the usual excess entropy production, which can appear only transiently and is called nonadiabatic. Another one is attributed solely to the breakage of detailed balance. The last part not satisfying the fluctuation theorem comes from the steady-state distribution asymmetry for odd-parity variables, which is activated in a non-transient manner. The latter two contributions combine together as the house-keeping (adiabatic) entropy production, whose positivity is not guaranteed except when the excess entropy production completely vanishes. where P r is the probability of a sequence r. As a corollary, the Jensen's inequality guarantees R ≥ 0. Consider r as a path or trajectory in state space, generated during a time interval by a stochastic dynamics. In case when its functional R [7] represents the total entropy production during the process, the FT has been derived for various nonequilibrium(NEQ) processes, and the thermodynamic 2nd law ∆S tot ≥ 0 automatically follows [3,4,8].More recently, Hatano and Sasa found that a part of the total entropy production (excess entropy), ∆S ex , also satisfies the FT, which represents the entropy production associated with transitions between steady states [9,10]. Later, Speck and Seifert showed that the remaining part (house-keeping entropy), ∆S hk , also satisfies the FT, which is required to maintain the NEQ steady state (NESS) [11,12]. In case of (quasi-static) reversible processes, the system stays at equilibrium almost always during the process, then the house-keeping entropy production vanishes, ∆S eq hk = 0. Most recently, Esposito et. al.[6] interpreted the house-keeping entropy as an adiabatic part and the excess entropy as a nonadiabatic part of the total entropy production, through a time-scale argument.Most of findings about the FTs so far hold only when all state variables have even parity under time reversal, such as position variables. A typical example is the driven Brownian motion in the over-damped limit. Including odd-parity variables, such as momentum, the mathematical description becomes more complicated in particular for NEQ processes. Recently, Spinney and Ford suggested a separation of the total entropy production into three terms for the stochastic system with odd-parity variables [13]. The excess entropy production can be cleanly separated out (in fact, exactly the same as in the case with even-parity variables only) and it satisfies the FT. However, the house-keeping part composes of two different terms and only one term satisfies the FT. Especially, the other term not satisfying the FT turns out to be transient, which seems inconsistent with the usual adiabatic feature of the house-keeping entropy. Thus, it was concluded that the physical interpretation of separated entropies is not as clear as in the even-variable only case (adiab...

We investigate the stochastic motion of a Brownian particle in the harmonic potential with a time-dependent force constant. It may describe the motion of a colloidal particle in an optical trap where the potential well is formed by a time-dependent field. We use the path integral formalism to solve the Langevin equation and the associated Fokker-Planck (Kramers) equation. Rigorous relations are derived to generate the probability density function for the time-dependent nonequilibrium work production beyond the overdamped limit. We find that the work distribution exhibits an exponential tail with a power-law prefactor, accompanied by an interesting oscillatory feature (multiple pseudo-locking-unlocking transitions) due to the inertial effect. Some exactly solvable cases are discussed in the overdamped limit.

We investigate an unconventional nature of the entropy production (EP) in nonequilibrium systems with odd-parity variables that change signs under time reversal. We consider the Brownian motion of a particle in contact with a heat reservoir, where particle momentum is an odd-parity variable. In the presence of an external momentum-dependent force, the EP transferred to environment is found not equivalent to usual reservoir entropy change due to heat transfer. There appears an additional unconventional contribution to the EP, which is crucial for maintaining the non-negativity of the (average) total EP enforced by the thermodynamic second law. A few examples are considered to elucidate the novel nature of the EP. We also discuss detailed balance conditions with a momentum-dependent force.

Time-integrated quantities such as work and heat increase incessantly in time during nonequilibrium processes near steady states. In the long-time limit, the average values of work and heat become asymptotically equivalent to each other, since they only differ by a finite energy change in average. However, the fluctuation theorem (FT) for the heat is found not to hold with the equilibrium initial ensemble, while the FT for the work holds. This reveals an intriguing effect of everlasting initial memory stored in rare events. We revisit the problem of a Brownian particle in a harmonic potential dragged with a constant velocity, which is in contact with a thermal reservoir. The heat and work fluctuations are investigated with initial Boltzmann ensembles at temperatures generally different from the reservoir temperature. We find that, in the infinite-time limit, the FT for the work is fully recovered for arbitrary initial temperatures, while the heat fluctuations significantly deviate from the FT characteristics except for the infinite initial-temperature limit (a uniform initial ensemble). Furthermore, we succeed in calculating finite-time corrections to the heat and work distributions analytically, using the modified saddle point integral method recently developed by us. Interestingly, we find noncommutativity between the infinite-time limit and the infinite-initial-temperature limit for the probability distribution function (PDF) of the heat.

Conventional wisdom indicates that initial memory should decay away exponentially in time for general (noncritial) equilibration processes. In particular, time-integrated quantities such as heat are presumed to lose initial memory in a sufficiently long-time limit. However, we show that the large deviation function of time-integrated quantities may exhibit initial memory effect even in the infinite-time limit, if the system is initially prepared sufficiently far away from equilibrium. For a Brownian particle dynamics, as an example, we found a sharp finite threshold rigorously, beyond which the corresponding large deviation function contains everlasting initial memory. The physical origin for this phenomenon is explored with an intuitive argument and also from a toy model analysis. Hot coffee gets colder and iced coffee gets warmer at room temperatures. These phenomena are the examples of equilibration processes and can be generalized as the following situation; a system with initial temperature T s is in thermal contact with a heat bath with temperature T b . Then, the system gradually deviates from its initial state and approaches to the final equilibrium (EQ) state which is determined by the heat bath. Here, the initial distance from final equilibrium is parameterized by the temperature ratio β ≡ T b /T s . The relaxation process is usually exponentially fast, so the memory of the initial temperature will be lost for average values of most physical observables after a characteristic relaxation time. However, the initial memory can often survive in the tail part (rare-event region) of a probability distribution function (PDF) even in the long-time limit.What about time-integrated quantities such as heat, work, or entropy production, which are the key quantities for nonequilibrium (NEQ) fluctuation theorems [1][2][3][4][5]? These accumulated quantities are also affected by a finite transient period, but their average values increase (or decrease) linearly in time asymptotically in NEQ steady state. Therefore, in a sufficiently long-time limit, our conventional wisdom may lead us to expect that they will lose all initial memory (independent of β). Nevertheless, in this Letter, we show rigorously that this is false wisdom for time-integrated quantities and, in particular, corresponding large deviation functions depend strongly on the initial condition (β) even in the infinite-time limit. More surprisingly, there exists a sharp threshold for β −1 in general, only beyond which the initial memory lasts forever.In literatures, there have been some reports that initial conditions can affect the large deviation function in the long time limit [6][7][8][9]. For example, van Zon and Cohen [6] showed that heat transfer Q in a driven harmonic oscillator in contact with a heat bath violates the fluctuation theorem even in the long-time limit, starting initially from EQ. In contrast to work, heat is known to satisfy the fluctuation theorem, only starting with a uniform distribution (infinite-temperature initial state) [10]. ...