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].

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

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-

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 푥푦 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.

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]

푃퐴 = 푁퐾 (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. The system boundary is diffusive, where the reflected particle direction is random,

and the reflected particle velocity (푣) follows the 2D Maxwell-Boltzmann distribution (푝(푣)) with a constant 퐾 =푘푇. The simulation box is divided into two areas by a narrow circular band: the surrounding plain and the central plateau. The circular band is the transition step, in which the local dimension is denoted by 푧, along the radius direction toward the center. No long-range force is applied on the particles on the plain and the plateau. The particles in the transition step are subject to a constant force, 푚푔, along −푧. The width of the transition step is 푧̂.

Figure 3. The Monte Carlo simulation of billiard-like particles freely moving on the surrounding “plain” and the central “plateau”, across the transition step. The dashed circles indicate the boundaries of the transition step with the outer plain and the central plateau.

Figure 4. Typical time profiles of the particle density ratio (휌), with the initial 휌 being (A) 0.60, (B) 1.00, and (C) 1.40. The steady-state particle density ratios are respectively 0.59, 0.59, and 0.60, unrelated to the initial condition. The parameter setting is similar with Case R1 in Figure 6, except that the gravitational acceleration (푔) is two times smaller.

For each simulation case, at each time step, the particle numbers on the plain (푁) and on

the plateau (푁) are counted. The average particle density ratio 휌 = is computed for every 4 3 1~410 time steps; the lifting force (퐹) is calculated as the average 푚푔푁 for every 1~2.510

time steps; the in-plane pressure (푃) is obtained as (∑ ∆푝̅)⁄ (∆푡̅ ∙퐿), where “” indicates 3 summation for all the particle-boundary collisions during ∆푡,̅ ∆푡 ̅ is 4~810 time steps, 퐿 is the length of the system boundary, and ∆푝̅ is the change in particle momentum in the normal direction. Initially, the particles are evenly placed on the plain and the plateau; the particle velocity follows 푝(푣) and the direction is random. Each simulation case is continued until the steady state of 푃, 퐹, and 휌 has been reached. The initial 휌 does not affect the steady state (Figure 4). At the steady state, there is no overall heat exchange with the environment (Figure 5).

Figure 5. Time profiles of the root mean square velocity (푣̅) of the incident and the reflected particles at the system boundary. The parameter setting is the same as Case R1 in Figure 6.

3. Results of the Monte Carlo Simulation

3.1 Non-Boltzmann particle distribution

Figure 6 shows the steady-state particle density ratio () as a function of the 푧̂⁄ ratio. For different simulation cases, the nominal mean free path of the particles,  ≈ [11], is √ adjusted by changing the particle diameter (푑) and the plain area (퐴). The value of 푧̂ is also varied, and 푔 is controlled to keep 푚푔푧̂⁄ 퐾 = 0.5, so that the Boltzmann factor ( = exp(−푚푔푧̂⁄ 퐾 )) is constant 0.607. Table 1 lists the parameter setting. The simulation setup is scalable; an example of the unit system can be based on Å, fs, g/mol, and K. The total particle number is 푁 = 800; 푇 = 1000, and 퐾 =8.3142 × 10; 푚 =1; the simulation time step is Δ푡 =1; the normalization factor of the particle diameter (푑) is 푑 = 1.6; the plateau area is 퐴 =1.256 × 10 ; the normalization factor of 푧̂ is 푧̂ = 10; the

normalization factor of 푔 is 푔 =4.16 × 10 . The error bars are the confidence interval, ± ∙

, where 푛 is the number of data points, 푠 is the standard deviation, and  is the inverse of Student’s t-distribution with the confidence level of 90%.

Figure 6. The steady-state particle density ratio (휌) as a function of the 푧̂⁄ ratio. The simulation case numbers (R1 to R7) are shown. It can be seen that the 푧̂⁄ ratio significantly affects 휌: 휌 is ̂ close to the Boltzmann factor (훿 =푒 ) when 푧̂⁄ is relatively large (i.e., when the system is chaotic), and close to 훿 when 푧̂⁄ is small (i.e., when the transition step is locally nonchaotic).

Table 1 Parameter setting for the simulation cases in Figure 6

Case Number 풛/흀퐅 푨퐏/푨퐆 풅⁄풅ퟎ 풛/풛ퟎ 품/품ퟎ R1 0.1006 1.8 1.0000 1.0 1.0000 R2 0.4110 1.8 1.2500 3.6 0.2778 R3 1.6069 1.8 2.2375 10.0 0.1000 R4 2.1966 1.8 2.8125 11.5 0.0870 R5 2.8077 1.8 3.3750 12.9 0.0775 R6 6.0179 3.2 7.5000 20.0 0.0500 R7 6.8617 4.8 8.7500 30.0 0.0333

When 푧̂⁄ ≈ 0.1, the transition step is nonchaotic, and  is close to  = 0.368; i.e., the steady state is nonequilibrium. It should be attributed to that, without extensive particle collision in the transition step, the particle motion along 푧 is dominated by the 푧-dimension momentum.

The trend is clear that  increases with 푧̂⁄, especially in the range of 푧̂⁄ from 1 to 4. When

푧̂⁄ is relatively large, the transition step is chaotic, and  converges to  = 0.607; i.e., the system reaches thermodynamic equilibrium.

3.2 Produced work in an isothermal cycle

Figure 7 shows an isothermal operation cycle. At State I, 푧̂⁄푧 = 0.25, 퐴⁄퐴 = 0.888, and 푧̂⁄ ≈ 0.1, where 푧 = 퐾⁄(푚푔) is used as the characteristic step width. From State I to II,

퐴 is constant and 푧̂⁄푧 increases to 0.5. As 푧̂ rises, less particles are on the upper plateau, so that

퐹 decreases while 푃 becomes larger. From State II to III, 푧̂ is constant and 퐴⁄퐴 expands to

1.764. Since the particle density is reduced, both 퐹 and 푃 are smaller. From State III to IV, 퐴 does not vary and 푧̂⁄푧 is lowered back to 0.25. Because the energy barrier of the transition step is less, 퐹 increases and 푃 decreases. Finally, the system returns to State I, and the densification of the particles leads to the increase in both 퐹 and 푃. Table 2 lists 푧̂⁄푧, 퐴⁄퐴, as well as the steady-state 퐹, 푃, and 휌 for each simulated system state. In addition to States I, II, III, and IV, there are 12 intermediate states in between them, marked as I-a, I-b, etc. State I is the same as Case

R1 in Figure 6. The error bars are calculated as the confidence interval, ± ∙ .

Figure 7. In an isothermal cycle, the system evolves from State I ([푧̂⁄푧, 퐴⁄퐴] = [0.250, 0.888]) to II ([0.500, 0.888]), III ([0.500, 1.764]), IV ([0.250, 1.764]), and back to State I. State I is the same as Case R1 in Figure 6 (푧̂⁄ ≈ 0.1). The solid lines are the regression curves from Equations (5) and (6). The work produced by 푃 (푊) is greater than the work consumed by 퐹 (푊): 푊⁄푊 = 1.497. The normalization factors are 퐹 = 푚푔푁, 푧 = 퐾⁄푚푔, and 푃 = 푁퐾⁄퐴.

Table 2 State evolution of the isothermal cycle in Figure 7

System state 풛⁄풛ퟎ 푨퐏⁄푨퐆 푭퐆⁄푭퐆ퟎ 푷⁄푷ퟎ 흆 I 0.2500 0.8879 0.3843 0.4417 0.5932 I-a 0.3125 0.8879 0.3521 0.4453 0.5066 I-b 0.3750 0.8879 0.3273 0.4536 0.4758 I-c 0.4375 0.8879 0.2929 0.4620 0.4056 II 0.5000 0.8879 0.2723 0.5179 0.3556

II-a 0.5000 1.1069 0.2480 0.4361 0.3859 II-b 0.5000 1.3258 0.2200 0.3871 0.3924 II-c 0.5000 1.5448 0.1893 0.3462 0.3754 III 0.5000 1.7637 0.1679 0.3032 0.3683 III-a 0.4375 1.7637 0.1898 0.2817 0.4404 III-b 0.3750 1.7637 0.2076 0.2803 0.4476 III-c 0.3125 1.7637 0.2317 0.2789 0.5522 IV 0.2500 1.7637 0.2478 0.2739 0.6340 IV-a 0.2500 1.5448 0.2689 0.3006 0.6235 IV-b 0.2500 1.3258 0.2991 0.3403 0.6155 IV-c 0.2500 1.1069 0.3245 0.3729 0.5840

From State I to II (through States I-a, I-b, and I-c) and from State III to IV (through States

III-a, III-b, III-c), the plain-to-plateau area ratio (퐴⁄퐴) remains constant. As 푧̂ varies, the particle density ratio (휌) changes. In accordance with Equation (2), the lifting force is 퐹 = 푚푔푁 ≈ (3)

The simulation data of 휌 approximately fit with

̂ 휌 = exp −훼 (4) where 훼 captures the effect of 푧̂⁄. The requirement of local equilibrium demands that 훼 must be a constant independent of location [8]. If 훼 =1, the right-hand side of Equation (4) is reduced to the Boltzmann factor. When 푧̂⁄ ≪1, because the crossing ratio () is less than , 훼 is generally larger than 1. Substitution of Equation (4) into Equation (3) gives 퐹 = (5) In Equation (5), 훼 is the only adjustable parameter for data fitting; all the other parameters are known. When 훼 is set to 2.28, Equation (5) agrees well with the MC simulation data of the 퐹 − 푧̂ relationship from State I to II; when 훼 is set to 2.06, Equation (5) agrees well with the simulation data from State III to IV, as shown by the upper and lower solid curves in Figure 7(A). From State II to III (through II-a, II-b, and II-c) and from State IV to I (through IV-a, IV- b, and IV-c), 푧̂ is kept constant while the plain area (퐴) changes. Based on Equation (1), the simulated in-plane pressure may be expressed as

푃= ≈ (6)  where  is the average particle density ratio and 퐾 = 푚푣̅ = 5.4459 × 10 is calculated from a reference system with 푧̂ =0 (see Figure 8). The upper and lower solid curves in Figure 7(B) are from Equation (6). For the upper curve,  is 0.3755, the average 휌 of States II, II-a, II-b, II-c, and III; for the lower curve,  is 0.6100, the average 휌 of States IV, IV-a, IV-b, IV-c, and I. It can be seen that Equation (6) is in agreement with the simulation result of the 푃−퐴 relationship.

Figure 8. Typical time profile of the average particle velocity (푣̅) in a reference system with 푧̂ = 0. The parameter setting is similar to Case R1 in Figure 6.

The 퐹 − 푧̂ loop consumes work 푊 =− ∫ 퐹d푧̂ + ∫ 퐹d푧̂ = 23.2퐾, calculated as the area enclosed by the upper and lower 퐹 − 푧̂ curves in Figure 7(A). The 푃−퐴 loop produces work 푊 = ∫ 푃d퐴 − ∫ 푃d퐴 = 34.7퐾, calculated as the area in between the upper and lower 푃−퐴 curves in Figure 7(B). The ratio between 푊 and 푊 is 푊⁄푊 =1.497. After a complete cycle, the overall work production is 푊 =푊 −푊 = 11.5퐾.

4. Discussion

4.1 Cross-influence of thermally correlated thermodynamic driving forces

The setting of the MC simulation is similar to an ideal-gas system. The only thermal reservoir is the environment, at constant 퐾. With the single heat source/sink, the Kelvin-Planck statement of the second law of thermodynamics dictates that no useful work can be produced in a cycle [12], which demands the symmetry in the cross-influence of thermally correlated 퐹 and 퐹.

Denote entropy by 푆 and heat capacity by 퐶. Based on =− and =− [8], we

have =− + =− + . Since = and = [8], it can be

obtained that = , i.e., 퐹 = 퐹 (7) which may be viewed as the counterpart of the Onsager relations [13].

Figure 9. The Kelvin-Planck statement of the second law of thermodynamics dictates that the cross-influence of the thermally correlated thermodynamic driving forces (퐹 and 퐹) must be symmetric (Equation 7). Indexes I-IV indicate the system states in an isothermal cycle; 푥 and 푥 are the conjugate variables of 퐹 and 퐹, respectively; d푥 and d푥 can be arbitrarily small.

Equation (7) is illustrated by the isothermal cycle in Figure 9. Two thermodynamic driving forces (퐹 and 퐹) are applied on a system with arbitrary ergodicity and chaoticity. With an arbitrarily small increment of d푥 or d푥, the work that 퐹 or 퐹 does to the system is 퐹d푥 or 퐹d푥, respectively. Without losing generality, we assume that 퐹 and 퐹 are negative. From State I to II, 푥 is constant; 푥 increases by d푥, and the cross-influence causes 퐹 to vary by 퐹d푥. From

State II to III, 푥 is constant; 푥 decreases by −d푥, and the cross-influence causes 퐹 to vary by −퐹d푥. From State III to IV, 푥 does not vary; 푥 decreases by −d푥 and correspondingly, 퐹 varies by −퐹d푥. From State IV to I, the system returns to the initial state; 푥 rises by d푥, and 퐹 is changed by 퐹d푥. The overall work done by 퐹 to the environment is 푊 = 퐹d푥d푥; the overall work done by 퐹 is 푊 =−퐹d푥d푥. The 퐹 − 푥 loop results in a heat loss ∆푞; the 퐹 −

푥 loop causes a heat absorption ∆푞. The Kelvin-Planck statement of the second law of

thermodynamics claims that after a complete cycle, ∆푞 +∆푞 =0 and 푊 + 푊 =0 [14], both of which lead to Equation (7).

4.2 Nonequilibrium particle density ratio

For the model system in Figure 1(B), the two thermodynamic driving forces are 푃 and 퐹;

퐹 is , and 퐹 is − . Thus, Equation (7) becomes ̂ =− (8) ̂

Substitution of Equations (1) and (2) into Equation (8) gives = , to satisfy which ̂ ̂ needs to be 푓 exp , with 푓 being a certain differentiable function. When 푧̂ =0, = 푓 = . Therefore, 푓() = , and the solution of Equation (8) is 

= = (9) ̂ where 퐴 = 퐴exp − . That is, the balance of the cross-influence of 푃 and 퐹 (Equation 8) is ̂ equivalent to that  equals to the Boltzmann factor, 훿 = exp − ; i.e.,   = = 푒̂ (10) 

where 휌 = and 휌 = . It is worth noting that Equation (10) is derived from Equation (8) and the system governing equations (Equations 1 and 2), not directly through the Lagrange multiplier method [e.g., 7]; the local chaoticity of the SND plays no role.

However, the MC simulation shows clearly that  is considerably influenced by the 푧̂⁄ ratio (Figure 6). Because Equation (10) is the solution of Equation (8), if  ≠ 훿, Equation (7) and Equation (8) cannot be satisfied, and the cross-influence of the two thermally correlated

thermodynamic driving forces (푃 and 퐹) would be asymmetric; that is, ≠− , or ̂ 퐹 ≠ 퐹 (11)

The mechanism of  ≠ 훿 is related to the crossing ratio of the particles (훿). Similar to the example in Figure 1(A), when 푧̂⁄ ≪1, the relevant kinetic energy for the particles to overcome the gravitational energy barrier from the plain to the plateau is mostly determined by the 푧-

dimension momentum, in average less than the total particle momentum by a factor of 2. Consequently,  → .

4.3 Production of useful work in an isothermal cycle

Inequality (11) causes the overall work production shown in Figure 7. According to 휌 =

 ∙ 휌 and 푁 = 푁 + 푁, we have 푁 = 푁⁄[1+ 퐴⁄퐴 ] and 푁 = 푁⁄[1+ 퐴⁄(퐴) ]. Based   on Equations (1) and (4), 푊 = 퐾 ∫ d퐴 − ∫ d퐴 = 푁퐾 ∙ ln , where  

퐴 and 퐴 are the compressed and the expanded plain areas, respectively;  and  are the particle density ratios associated with the lowered and the raised plateau, respectively; the integral bounds are the system states. Likewise, based on Equations (2) and (4), 푊 = 푚푔 ∫ 푁d푧̂ −   ⁄ ∫ 푁d푧̂ = ∙ ln . Thus, 푊 푊 = 훼.  

If 푧̂⁄ ≫1, the transition step is chaotic and  =  (i.e., 훼 =1), so that 푊 = 푊, in agreement with Equation (8). If 푧̂⁄ ≪1, the transition step is a locally nonchaotic SND and  →  (i.e., 훼 →2), so that

푊⁄푊 →2 (12)

In Section 3.2, the numerical result of the 푊⁄푊 ratio is 1.497, smaller than the ideal- case scenario of Equation (12). The difference should be attributed to the occasional particle collisions in the transition step in the MC simulation, the finite size and the local curvature of the transition step, the boundary effect of the relatively small simulation box, and the occupied volume of the particles. These factors also render the simulated 훼 different from 2.

4.4 Variants of the system setup

Figure 10(A) depicts one variant of the plateau-plain system. The right-hand side of the plateau boundary is connected to the lower plain by a narrow vertical step, and the left-hand side is connected through a wide ramp. The height of the transition step (푧̂) is much less than , while the ramp width (퐿) is much larger than . Therefore, the particle behavior is nonchaotic in the transition step, but chaotic in the ramp. Because the particle density ratios across the step and the

ramp tend to be different, there would be a spontaneous particle flow, which may continuously produce useful work by absorbing thermal energy from the environment.

Figure 10. (A) Side view of a system with an asymmetric plateau-plain border. Similar to Figure 1(B), the upper plateau is separated from the lower plain by an energy barrier. Different from

Figure 1(B), one side of the plateau-plain border is a wide ramp, where 퐿 ≫ ; the other side is a narrow transition step, where 푧̂ ≪ . (B) Another variant of the system. The plateau and the plain are connected through a chaotic wide ramp. There are nonchaotic traps on the plateau.

In Figure 10(A), if the step and the ramp are alternately closed and reopened (i.e., the plain- plateau border is switchable), the system may shift between the equilibrium state and the nonequilibrium steady state, without having an energetic penalty. When the ramp is closed by a frictionless sliding door and the transition step is open, similar to Figure 1(B), the system is in a nonequilibrium state and the particle density ratio (휌) is less than the Boltzmann factor (훿). As the plain area expands from 퐴 to 퐴, the system does work 푊 to the environment. Then, the ramp is opened and the transition step is closed by another frictionless sliding door. Since the entire system becomes ergodic and chaotic, 휌 equals to 훿. As the plain area is reduced from 퐴 back to 퐴, because the equilibrium 푁 is less than the nonequilibrium 푁, the work that the environment does to the system (푊) is smaller than 푊. After such a cycle, a useful work (푊) is generated, equal to the thermal energy absorbed from the environment. In accordance with 푁 =

푁⁄[1+ 퐴⁄퐴 ], 푊 = 푊 − 푊 = ∙∆ ln , where ∆() indicates the difference between the equilibrium state and the nonequilibrium state. There can be a variety of ways to arrange the nonchaotic component(s). For instance, in Figure 10(B), the entire plateau-plain border is a chaotic wide ramp, while there are a number of nonchaotic traps on the plateau. Particles randomly move into and out of the traps. The trap bottom

is a perfect heat exchanger at constant T, and does not move; i.e., the trap depth changes with the plateau height (푧̂). The trapped particles do not contribute to the lifting force, 퐹. A useful work

(푊) may be produced if the traps are periodically closed and opened and the plain expands and shrinks accordingly, or the traps remain open while 푧̂ and 퐴 are alternately adjusted in an isothermal cycle similar to Figure 7.

4.5 Difference from Maxwell’s demon

The concept of SND is fundamentally different from Maxwell’s demon. The latter has been extensively studied for many decades [e.g., 15-19]. Maxwell’s demon is nonspontaneous. Its work production is counterbalanced by the energetic penalty of information processing [20-23]. For the SND-based system, no particle microstate is directly observed, and the physical nature of information is not involved.

4.6 Implications

The role of SND may be described by

퐹 = 퐹 +∆퐹 (13) where 퐹 is the Helmholtz free energy, 퐹 is the equilibrium Helmholtz free energy, ∆퐹 = 퐹 +

퐹 is the additional Helmholtz free energy caused by the SND, and 퐹 and 퐹 represent the effects of internal energy and entropy, respectively. For the model system under investigation, 퐹 = −∆퐸훿푁, where ∆퐸 = 푚푔푧̂, 훿푁 =∆ is the number of excess particles across the SND. ⁄

According to the equation of entropy of ideal gas [8], 퐹 =− ∆ 푁ln + 푁ln . The system efficiency may be measured by 푘 = (14) where 푄 is the total thermal energy stored in the system. It is worth noting that the nonequilibrium state does not necessarily cause a contradiction to Equation (7). A counterexample can be formed if both thermodynamic driving forces in the

model system are chosen as the in-plain pressure; i.e., 퐹 = 푃 and 퐹 = 푃, where 푃 is the in- plane pressure on the plateau. Thus, Equation (7) becomes = (15) Equation (15) does not meet Condition 2 in the last paragraph of the introductory section, since the central and the surrounding areas are governed by the same equation of state (Equation 1). It

∗ ∗ leads to −퐴 = 퐴 , which has the general solution 푁 = 푓 , with 푓 being a

differentiable function. Because 푁 = 푁1+ , Equation (15) is satisfied. That is, the cross-  influence of 푃 and 푃 is always symmetric, no matter whether  follows the Maxwell-Boltzmann distribution or a non-Boltzmann distribution. In order to obtain an appreciable nonequilibrium effect in Figure 1(B), the gravitational field must be ultra-strong, at the level of neutron stars or small black holes. In addition to gravity, other thermodynamic driving forces that may be relevant to SND include inertial force, gas or plasma pressure, chemical potential, Coulomb force, magnetic moment, angular momentum, entropy-related forces in condensed matters (e.g., osmotic pressure and degeneracy pressure), surface and interface tension, among others. It would be interesting to explore whether SND-like mechanisms may exist in nature, from small time and length scales (such as superstring theory and quantum mechanics), to intermediate scales (such as molecular engineering, condensed matters, gas or plasma systems, and life science), to the cosmological scale (e.g., large-푔 environments, the weak photon scattering in space, etc.), as well as information theory. Moreover, the short local event duration (푡̅ ) does not break the microscopic reversibility [24], but it influences the basic assumptions of the Boltzmann equation and the H-theorem [8]. These will be the important topics of the future study.

5. Concluding Remarks

In the current research, we investigate a billiard-type model system (Figure 1B), in which elastic particles randomly move across a locally nonchaotic energy barrier between an upper plateau and a lower plain. The energy barrier is a type of spontaneously nonequilibrium dimension, as its size is much smaller than the mean free path of the particles. It leads to a non-Boltzmann steady-state particle distribution, without any specific knowledge of the system microstate.

The second law of thermodynamics (Equation 7) dictates that the particles must follow the Maxwell-Boltzmann distribution (Equation 10); otherwise, useful work may be produced in a cycle by absorbing heat from a single thermal reservoir (e.g., Equation 12). However, our Monte Carlo simulation confirms that the particle distribution is indeed nonequilibrium (Figure 6). Without extensive particle collision in the transition step, there is no mechanism for the system to reach thermodynamic equilibrium. Consequently, in an isothermal cycle, the total produced work is greater than the total consumed work (Figure 7), contradicting the Kelvin-Planck statement. Such an anomalous phenomenon is attributed to the asymmetry in the cross-influence of the thermally correlated thermodynamic driving forces (Inequality 11). There are a number of variants that can achieve similar effects, e.g., with an asymmetric or switchable plateau-plain boundary (Figure 10A).

Appendix: Nomenclature

퐴: Area of the upper plateau

퐴 = 퐴 ∙ exp(− 푧̂⁄푧 ): Adjusted plateau area

퐴: Area of the lower plain

퐴 and 퐴: Expanded and compressed plain areas, respectively

퐶: Heat capacity 푑: Particle diameter

푑: Normalization factor of 푑

퐷: Plateau size 퐹: Helmholtz free energy

퐹: Helmholtz free energy at thermodynamic equilibrium

퐹: Lifting force on the plateau

퐹 = 푚푔푁: Normalization factor of 퐹

퐹: Component of 퐹 caused by the centrifugal force

퐹: Component of 퐹 caused by the particle weight

퐹 and 퐹: Thermodynamic driving forces in area 푖 and area 푗, respectively

퐹 = : Cross-influence of 푥 on 퐹

퐹: Additional Helmholtz free energy related to internal energy

퐹: Additional Helmholtz free energy related to entropy 푔: Gravitational acceleration

푔: Normalization factor of 푔 i and j: Two large ergodic and chaotic areas I, II, III, and IV: System states 푘: System efficiency

푘: Boltzmann constant K: Particle kinetic energy 퐾: Average particle kinetic energy

퐾: Characteristic 퐾 in the reference system

퐾 = 푚푣: Characteristic kinetic energy

퐿: System size

퐿: Circumference of the plateau 퐿: Length of the chaotic ramp 푚: Particle mass

푛: Number of the data points 푁: Total particle number

푁 and 푁: Numbers of the particles on the plateau and the plain, respectively 푝: Probability density of the particle velocity

푝: Probability density of the 푧-dimension particle velocity component 푃: In-plane pressure of the plain

푃 = : Normalization factor of 푃

푃: In-plane pressure of the plateau

푞 : Absorbed or released heat from State 푖 to State 푖 (푖, 푖 = 1,2,3,4)

푄: total thermal energy stored in the system

푠: Standard deviation 푆: Entropy

푡̅ : Characteristic interaction time

푡̅ : Characteristic event duration T: Temperature 푣: Particle velocity 푣̅: Average particle velocity

푣̅ : Root mean square particle velocity

푣: 푧-dimension particle velocity component

푣̅ : Average 푧-dimension particle velocity component

푣: Characteristic 푣 of the particles changing direction in the transition step

푊 and 푊: Works done and received by the plain, respectively

푊: Work consumed by 퐹

푊 and 푊: Work produced/consumed by 퐹 and 퐹, respectively

푊 : Produced or consumed work from State 푖 to 푖

푊: Work produced by 푃

푊: Overall produced work 푥: Horizontal dimension

푥 and 푥: Conjugate variables of 퐹 and 퐹, respectively

푥̅: Average value 푦: Horizontal dimension 푧: Vertical dimension 푧̂: Height of the transition step or vertical plane

푧̂ : Normalization factor of 푧̂

푧 = 퐾⁄(푚푔): Characteristic height 훼: Parameter of the non-Boltzmann distribution 훽 = : Thermodynamic beta

 = ⁄: Crossing ratio

: Boltzmann factor

: Probability for the particles to move from area i to j

: Probability for the particles to move from area j to i 훿푁: Number of the excess particles across the barrier

: Inverse of Student’s t-distribution ∆(): Difference between the equilibrium state and the nonequilibrium state ∆퐸: Potential difference between areas i and j ∆퐹: Additional Helmholtz free energy caused by the SND

∆푝̅ : Change in momentum along the normal direction

∆푞: Heat loss caused by 퐹

∆푞: Heat absorption caused by 퐹

∆푞: Absorbed heat (particle kinetic energy)

∆푡: Time step in the computer simulation ∆푡:̅ Number of timesteps for the calculation of average value ∆푡̃: Characteristic time for the particles to change direction in the transition step

∆푈: Internal energy change from State 푖 to State 푖 (푖, 푖 = 1,2,3,4)

∆: Excess chemical potential in the upper plateau

∆: Excess chemical potential in the lower plain

: Nominal mean free path of the particles : Particle density ratio : Average value of 

휌: Particle density at the bottom of the plane

휌 = 푁⁄퐴: Particle density on the plateau

 and : Particle densities in area 푖 and area 푗, respectively

 and :  associated with lowered and raised 푧̂, respectively

휌 = 푁⁄퐴: Particle density in the plain

References

1. B. Gyenis. Maxwell and the normal distribution: A colored story of probability, independence, and tendency toward equilibrium. Studies History Phil. Modern Phys. 57, 53- 65 (2017). 2. A. Klenke. Probability Theory (Springer Berlin Heidelberg, 2013) 3. J. L. Lebowitz, O. Penrose. Modern ergodic theory. Phys. Today 26, 23-29 (1973) 4. E. Lutz, F. Renzoni. Beyond Boltzmann-Gibbs statistical mechanics in optical lattices. Nat. Phys. 9, 615-619 (2013). 5. A. Argun, A. Moradi, E. Pince, G. B. Bagci, A. Imparato, G. Volpe. Non-Boltzmann stationary distributions and nonequilibrium relations in active baths. Phys. Rev. E 94, 062150 (2016). 6. G. Lebon, D. Jou. Understanding Nonequilibrium Thermodynamics (Springer, 2008) 7. R. K. Pathria, P. D. Beale. Statistical Mechanics (Elsevier, 2011) 8. M. Kardar. Statistical Physics of Particles (Cambridge Univ. Press, 2007) 9. L. Han. Ball Collision Simulation. https://www.mathworks.com/matlabcentral/ fileexchange/41032-ball-collision-simulation (2015) 10. Z. Shang, Y. Qiao. Matlab Program for the Billiard-Type Model System. http://mmrl.ucsd.edu/Z_Upload/papers/DragonsEgg.zip (2019) 11. H. Kuhn, H. D. Forsterling, D. H. Waldeck. Principles of Physical Chemistry (John Wiley & Sons, 2009). 12. H. Struchtrup. Thermodynamics and Energy Conversion (Springer New York, 2014). 13. S. R. De Groot, P. Mazur. Nonequilibrium Thermodynamics (Dover, 1984). 14. K. Huang. Statistical Mechanics (John Wiley & Sons, 1987) 15. T. Sagawa. Thermodynamics of Information Processing in Small Systems (Springer Japan, 2013). 16. A. L. P. Livorati, A. Loskutov, E. D. Leonel. A peculiar Maxwell’s demon observed in a time-dependent stadium-like billiard. Physica A 391, 4756-4762 (2012). 17. J. V. Koski, V. F. Maisi, T. Sagawa, J. P. Pekola. Experimental observation of the role of mutual information in the nonequilibrium dynamics of a Maxwell demon. Phys. Rev. Lett. 113, 030601 (2014).

18. J. V. Koski, A. Kutvonen, T. Ala-Nissila, J. P. Pekola. On-chip Maxwell’s demon as an information-powered refrigerator. Phys. Rev. Lett. 115, 260602 (2015). 19. K. Chida, S. Desai, K. Nishiguchi, A. Fujiwara. Power generator driven by Maxwell’s demon. Nature Commun. DOI: 10.1038/ncomms15301 (2017). 20. R. Landauer. Information is inevitably physical, in Feynman and Computation, pp77-92 (Perseus Books, Cambridge, MA, 1998) 21. C. H. Bennett. Notes on Landauer’s principle, reversible computation and Maxwell’s demon, Studies in History and Philosophy of Modern Physics 34, 501–510 (2003) 22. D. Mandal, C. Jarzynski. Work and information processing in a solvable model of Maxwell’s demon. Proc. Natl. Acad. Sci. U.S.A. 109, 11641 (2012) 23. J. M. R. Parrondo, J. M. Horowitz, T. Sagawa. Thermodynamics of information. Nat. Phys. 11, 131 (2015). 24. N.G. Van Kampen. Stochastic Processes in Physics and Chemistry (Elsevier, 2007)