具有滞后和亚稳态的非平衡热力学过程--Legendre子流形上具有不稳定段和稳定段的接触哈密顿量
Nonequilibrium thermodynamic process with hysteresis and metastable states -- A contact Hamiltonian with unstable and stable segments of a Legendre submanifold
摘要：
本文在接触流形上描述了自旋统计热力学系统相变的动力学过程，这个动力学过程是从亚稳态平衡态演化到最稳定的对称破缺平衡态的过程。假设对称破缺相或有序相的亚稳和最稳定的平衡态可以用接触流形的勒让德子流形的修剪投影来描述，这些修剪投影表示滞后和伪自由能曲线.与相变相关的奇异性在这个框架中自然产生，正如勒让德奇异性理论所建议的那样。然后提出了一个特殊的切触Hamilton向量场，使得投影Legendre子流形的一个修剪段是切触流形的一个区域中的稳定不动点集，而另一个修剪段是不稳定不动点集.该接触哈密顿向量场被识别为从亚稳态平衡态到最稳定平衡态的动力学过程。为了明确地说明以上的陈述，我们着重讨论了具有长程相互作用的Ising型自旋模型，即Husimi-Temperley模型，其中该模型表现出相变。
英文摘要：
In this paper, a dynamical process in a statistical thermodynamic system of spins exhibiting a phase transition is described on a contact manifold, where such a dynamical process is a process that a metastable equilibrium state evolves into the most stable symmetry broken equilibrium state. Metastable and the most stable equilibrium states in the symmetry broken phase or ordered phase are assumed to be described as pruned projections of Legendre submanifolds of contact manifolds, where these pruned projections of the submanifolds express hysteresis and pseudo-free energy curves. Singularities associated with phase transitions are naturally arose in this framework as has been suggested by Legendre singularity theory. Then a particular contact Hamiltonian vector field is proposed so that a pruned segment of the projected Legendre submanifold is a stable fixed point set in a region of a contact manifold, and that another pruned segment is a unstable fixed point set. This contact Hamiltonian vector field is identified with a dynamical process departing from a metastable equilibrium state to the most stable equilibrium one. To show the statements above explicitly an Ising type spin model with long-range interactions, called the Husimi-Temperley model, is focused, where this model exhibits a phase transition.
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