稀气体非平衡定态的指数近似及其性质
Exponential approach to, and properties of, a non-equilibrium steady state in a dilute gas
摘要：
本文研究了在不同温度下与几个热库接触的系统的动力学模型$T_α$。我们的系统是一个空间均匀的稀气体，其内部动力学描述的非线性玻尔兹曼方程与麦克斯韦碰撞。类似地，与储层$α$的相互作用由具有麦克斯韦$M_{T_α}$作为其定态的马尔可夫过程表示。我们证明了非平衡稳态（NESS）的存在性和唯一性，并显示指数收敛到这个NESS的概率措施引入到麦克斯韦碰撞的研究Gabetta，Toscani和Wenberg（GTW）的度量。这表明当前速度分布与稳态速度分布之间的GTW距离是系统的李雅普诺夫泛函。我们还推导出表达式的熵产生系统加上水库，这是总是积极的。
英文摘要：
We investigate a kinetic model of a system in contact with several thermal reservoirs at different temperatures $T_α$. Our system is a spatially uniform dilute gas whose internal dynamics is described by the nonlinear Boltzmann equation with Maxwellian collisions. Similarly, the interaction with reservoir $α$ is represented by a Markovian process that has the Maxwellian $M_{T_α}$ as its stationary state. We prove existence and uniqueness of a non-equilibrium steady state (NESS) and show exponential convergence to this NESS in a metric on probability measures introduced into the study of Maxwellian collisions by Gabetta, Toscani and Wenberg (GTW). This shows that the GTW distance between the current velocity distribution to the steady-state velocity distribution is a Lyapunov functional for the system. We also derive expressions for the entropy production in the system plus the reservoirs which is always positive.
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