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A Monte Carlo Simulation of Locally Nonchaotic Barrier

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A Monte Carlo Simulation of Locally Nonchaotic Barrier Production of Useful Work in an Isothermal Cycle: A Monte Carlo Simulation of a Locally Nonchaotic Energy Barrier Y. Qiao,1,2,* Z. Shang1 1 Program of Materials Science and Engineering, University of California – San Diego, La Jolla, CA 92093, U.S.A. 2 Department of Structural Engineering, University of California – San Diego, La Jolla, CA 92093-0085, U.S.A. * Corresponding author (Email: [email protected]; phone: 858-534-3388) Abstract: A Monte Carlo simulation is performed on a billiard-type model system, which contains a locally nonchaotic energy barrier. Without extensive particle collision across the energy barrier, the system steady state is nonequilibrium; that is, the particles follow a non-Boltzmann distribution. Remarkably, as the energy barrier varies in an isothermal cycle, the total produced work is greater than the total consumed work, because of the asymmetry in the cross-influence of the thermally correlated thermodynamic driving forces. Such a phenomenon cannot be explained by the second law of thermodynamics. Similar anomalous effects may be achieved if the barrier is switchable or asymmetric. In essence, the energy barrier is a spontaneously nonequilibrium dimension. It is fundamentally different from Maxwell’s demon, unrelated to the physical nature of information. KEYWORDS: Nonequilibrium; Nonchaotic; The second law of thermodynamics; Energy barrier; Monte Carlo simulation 1. Introduction Randomness is a fundamental concept in statistical mechanics [1,2]. An ergodic and chaotic system always reaches thermodynamic equilibrium, while a nonergodic or nonchaotic system may not [e.g., 3-5]. The latter should be analyzed in the framework of nonequilibrium stochastic thermodynamics [6]. 1 Figure 1. (A) In a vertical 푦 − 푧 plane in a gravitational field (푔), if 푧̂ ≪ , at the steady state the particle density does not follow the Maxwell-Boltzmann distribution. (B) The billiard-type model system, wherein elastic particles randomly move in the horizontal 푥 − 푦 dimension in between a upper “plateau” and a lower “plain”, across a narrow transition step. If 푧̂ ≪ , the steady-state ⁄ particle density ratio ( = ) is inherently nonequilibrium. One example is shown in Figure 1(A), in which billiard-like elastic particles randomly move in a vertical 푦 − 푧 plane in a gravitational field (푔). The 푦 dimension is infinitely large. The upper boundary (푧 = 푧̂) is a specular wall, with 푧̂ being the plane height. The lower boundary (푧 = 0) is a diffusive wall at constant temperature (푇), where the reflected particle velocity (푣) is governed by the two-dimensional (2D) Maxwell-Boltzmann distribution [7]: 푝(푣) = ∙ exp − , with 푝 being the probability density, 푚 the particle mass, 퐾 = 푘 푇 the average particle kinetic energy (퐾), and 푘 the Boltzmann constant. If the mean free path of the particles () is small compared to the plane height (푧̂), the system is ergodic and chaotic. At 푧 = 푧̂, the local particle density, , is equal to [8], where is the particle density ratio, is the local ̂ particle density at the bottom of the plane, =푒 is the Boltzmann factor, and 훽 = . If ≫ 푧̂, the characteristic interaction time 푡̅ = ⁄푣̅ is much longer than the characteristic event duration 푡̅ =2푧̂⁄푣̅ , where 푣̅ is the average particle velocity and 푣̅ is the average 푧-dimension particle velocity component (푣). On the time scale of 푡̅ , few particle collisions take place and thus, the particle trajectories tend to be nonchaotic. The probability for a particle to travel from 푧 =0 to 푧 = 푧̂ is mainly determined by 푣, relatively unrelated to the 푦- dimension momentum. As a first-order approximation, the particle density ratio at 푧̂ can be 2 assessed as = ∫ 푝 (푣 )d푣 = , where 푝 (푣 ) = exp − is the 2D Maxwell- ̂ Boltzmann distribution of 푣. Clearly, the nonchaotic is different from . Figure 2. (A) In a system containing a spontaneously nonequilibrium dimension (SND), the steady-state particle distribution is intrinsically in a non-Boltzmann form; i.e., ≠ . (B) In an ergodic and chaotic system, at thermodynamic equilibrium, = . Figure 2(A) depicts two large ergodic and chaotic areas (푖 and 푗) that are separated by a nonchaotic and/or nonergodic barrier. A large number of elastic particles randomly move in the system. The crossing ratio of the barrier, = ⁄, does not equal to the Boltzmann factor, 훿 = ∆ 푒 , where ∆퐸 is the potential difference between the two areas, and and are the probabilities for the particles to cross the barrier from area 푖 to 푗 and from area 푗 to 푖, respectively. Hereafter, such a barrier will be referred to as a spontaneously nonequilibrium dimension (SND). Without any specific knowledge of the system microstate, SND offers a mechanism to reach a ⁄ nonequilibrium steady state; i.e., ≠ 훿, where = , with and being the steady-state particle densities in areas 푖 and 푗, respectively. In comparison, Figure 2(B) shows the same two large areas connected through a regular open gate, across which 훿 = 훿; at thermodynamic equilibrium, = 훿 [8]. In the current study, we investigate a SND-based system that satisfies the following six conditions: 1) The system is divided by the SND into two large areas (푖 and 푗). 2) Areas 푖 and 푗 are dominated by two different thermodynamic driving forces (퐹 and 퐹), respectively. 3) 퐹 and 퐹 are thermally correlated through particle diffusion; the energy change associated with their cross- 3 influence (퐹 = and 퐹 = ) is from heat, where 푥 and 푥 are the conjugate variables of 퐹 and 퐹, respectively. The operations of 퐹 and 퐹 are 4) reversible and 5) independent of each other, and 6) do not rely on temperature variation. 2. Monte Carlo Simulation 2.1 The billiard-type model system Figure 1(B) shows a SND-based billiard-type model system. A large number of elastic particles randomly move in the horizontal 푥푦 dimension. A uniform gravitational field (푔) is along the out-of-plane direction, −푧. The central area is higher, which will be referred to as “plateau”. The surrounding lower area will be referred to as “plain”. The plateau and the plain are separated by a transition step, which imposes an energy barrier to the particle motion from the plain to the plateau. The plateau height (푧̂) can be changed by the lifting force on the plateau, 퐹. The total particle number 푁 = 푁 + 푁 is constant, with subscripts “P” and “G” indicating the plain and the plateau, respectively. The plain area (퐴) can be adjusted by moving the outer system boundary; the area of the plateau (퐴) is fixed. The thermodynamic driving forces under investigation are the lifting force (퐹) and the in-plane pressure of the plain (푃), with the conjugate variables being 푧̂ and −퐴, respectively. It is assumed that i) the particle motion is frictionless; ii) the changes of 푧̂ and 퐴 are reversible; iii) the transition step is smooth, i.e., as the particles move across it, no energy is dissipated; and iv) the environment is a perfect heat reservoir of a constant average particle kinetic energy (퐾 = 푘푇). Moreover, v) we choose to study a system wherein 푧̂ is much smaller than the plain/plateau size; if 푔 =0, 푧̂ has little influence on the particle distribution. The model system meets the six conditions listed in the last paragraph of the introductory section: The plateau and the plain are separated by the energy barrier; they are dominated by 퐹 and 푃, respectively; 퐹 is directional and has a different governing equation from 푃; a variation in 퐴 or 푧̂ would cause a particle redistribution across the transition step, resulting in an exchange of thermal energy with the environment; 퐴 and 푧̂ can be adjusted reversibly and independently, and 퐾 is maintained constant by the thermal bath. 4 When 푧̂ ≫ , the system is ergodic and chaotic. If 푧̂ ≪ , the transition step becomes a locally nonchaotic SND. Notice that the in-plane pressure (푃), the plain area (퐴), the lifting force (퐹), and the plateau height (푧̂) can be measured from the surface of the system. In the following discussion, the system state will be defined by these macroscopic variables; no specific microstate will be directly observed. 2.2 Governing equations With Condition (v) in Section 2.1, the model system may be viewed as two large areas (the plain and the plateau) bordering each other across the line of transition step. The in-plane pressure of the plain is governed by [7] 푃퐴 = 푁퐾 (1) The lifting force on the plateau (퐹) contains two components: the particle weight 퐹 = 푚푔푁, and the centrifugal force (퐹) caused by the particles changing direction in the transition step. Denote the characteristic time for a particle to change its direction in the transition step by ∆푡̃. At the steady state, during ∆푡̃, on average 퐿(푣∆푡̃) particles pass through the transition step, where 푣 represents the average 푧̃-dimension velocity component of these particles, and 퐿 is the plateau circumference. The average centrifugal force per particle is 푚푣⁄∆푡̃. Thus, 퐹 is a fraction of 퐹, where 퐾 = 푚푣 and 퐷 is the plateau size. When 퐷 is much larger than 푧 = 퐾⁄(푚푔), 퐹 is negligible compared to 퐹, since 퐾 is on the same scale as 퐾. Under this condition, the lifting force may be calculated as 퐹 = 푚푔푁 (2) 2.3 Setup for the Monte Carlo simulation Nonequilibrium stochastic processes are often analyzed through Monte Carlo (MC) simulation. The program code of our MC simulation is partly based on [9] and can be downloaded from [10]. The simulation setup is illustrated in Figure 3. A relatively large number of billiard-like elastic particles freely move in a square box in the 푥 − 푦 plane.
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