Entropy growth during free expansion of an ideal gas
Entropy growth during free expansion of an ideal gas
复制标题
理想气体自由膨胀过程中的熵增长
DOI：
--
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
J. Lebowitz
中科院分区：
文献类型：
--
作者：
Subhadip Chakraborti;A. Dhar;S. Goldstein;A. Kundu;J. Lebowitz
为了说明玻尔兹曼构造的熵函数是为宏观系统的微观状态定义的，我们在这里提出了一个简单的例子，自由膨胀的一维气体的非相互作用的点粒子。这种构造要求我们定义对应于宏观变量的宏观状态。我们通过指定单个粒子位置-速度空间{x，v}的矩形框中的粒子分数ΔxΔv来定义宏观状态M。我们验证了当粒子数很大时，非平衡系综的典型微观态的玻尔兹曼熵S B（t）与与该系综相关的粗粒度时间演化单粒子分布的吉布斯熵一致。S B（t）接近于给定初始状态的动力学演化的最大可能值。接近的速率取决于宏观状态定义中Δv的大小，当Δv → 0时，在任何固定时间t趋于零。令人惊讶的是，当时间与Δv成比例时，不同的曲线S B（t）塌陷：t τ/Δv。我们找到了S B（τ）在极限Δv → 0时的一个显式表达式。我们还考虑了一个不同的，更流体动力学，定义的宏观态S B（t）是单调增加，不像以前的一个有小的衰减振荡附近的最大值。我们的系统是非遍历的，非混沌和非相互作用的，因此，我们的研究结果表明，这些概念是不相关的，因为有时声称，观察宏观不可逆性和熵增加。相反，初始条件，典型性，大量和粗粒度的概念是重要的因素。我们通过大量的模拟和分析结果来证明这些想法。
To illustrate Boltzmann’s construction of an entropy function that is defined for a microstate of a macroscopic system, we present here the simple example of the free expansion of a one dimensional gas of non-interacting point particles. The construction requires one to define macrostates, corresponding to macroscopic variables. We define a macrostate M by specifying the fraction of particles in rectangular boxes ΔxΔv of the single particle position-velocity space {x, v}. We verify that when the number of particles is large the Boltzmann entropy, S B(t), of a typical microstate of a nonequilibrium ensemble coincides with the Gibbs entropy of the coarse-grained time-evolved one-particle distribution associated with this ensemble. S B(t) approaches its maximum possible value for the dynamical evolution of the given initial state. The rate of approach depends on the size of Δv in the definition of the macrostate, going to zero at any fixed time t when Δv → 0. Surprisingly the different curves S B(t) collapse when time is scaled with Δv as: t ∼ τ/Δv. We find an explicit expression for S B(τ) in the limit Δv → 0. We also consider a different, more hydrodynamical, definition of macrostates for which S B(t) is monotone increasing, unlike the previous one which has small decaying oscillations near its maximum value. Our system is non-ergodic, non-chaotic and non-interacting; our results thus illustrate that these concepts are not as relevant as sometimes claimed, for observing macroscopic irreversibility and entropy increase. Rather, the notions of initial conditions, typicality, large numbers and coarse-graining are the important factors. We demonstrate these ideas through extensive simulations as well as analytic results.