Kac模型的巨正则演化
Grand Canonical Evolution for the Kac Model
摘要：
研究了在固定温度和化学势条件下，随机碰撞粒子与无限大储层相互作用的模型。粒子之间的相互作用通过Kac主方程\cite{kac}建模。此外，颗粒可以离开系统朝向储存器或从储存器进入系统。该系统在温度$T=β^{-1}$和化学势$χ$下存在唯一的正则包绕态。我们表明，任何初始状态的收敛指数快速平衡计算的频谱间隙的发电机在一个合适的$L^2$空间和显示指数下降的相对熵相对于稳定状态。我们还展示了混沌的传播，从而证明了玻尔兹曼-卡茨型方程在无限系统极限下的粒子密度的有效性。
英文摘要：
We study a model of random colliding particles interacting with an infinite reservoir at fixed temperature and chemical potential. Interaction between the particles is modeled via a Kac master equation \cite{kac}. Moreover, particles can leave the system toward the reservoir or enter the system from the reservoir. The system admits a unique steady state given by the Grand Canonical Ensemble at temperature $T=β^{-1}$ and chemical potential $χ$. We show that any initial state converges exponentially fast to equilibrium by computing the spectral gap of the generator in a suitable $L^2$ space and by showing exponential decrease of the relative entropy with respect to the steady state. We also show propagation of chaos and thus the validity of a Boltzmann-Kac type equation for the particle density in the infinite system limit.
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