Recent Advances in Conservation-Dissipation Formalism for Irreversible Processes

The main purpose of this review is to summarize the recent advances of the Conservation-Dissipation Formalism (CDF), a new way for constructing both thermodynamically compatible and mathematically stable and well-posed models for irreversible processes. The contents include but are not restricted to the CDFs physical motivations, mathematical foundations, formulations of several classical models in mathematical physics from master equations and Fokker-Planck equations to Boltzmann equations and quasi-linear Maxwell equations, as well as novel applications in the fields of non-Fourier heat conduction, non-Newtonian viscoelastic fluids, wave propagation/transportation in geophysics and neural science, soft matter physics, textit{etc.} Connections with other popular theories in the field of non-equilibrium thermodynamics are examined too.

Hamiltonian Dynamics of Saturated Elongation in Amyloid Fiber Formation

Elongation is a fundament process in amyloid fiber growth, which is normally characterized by a linear relationship between the fiber elongation rate and the monomer concentration. However, in high concentration regions, a sub-linear dependence was often observed, which could be explained by a universal saturation mechanism. In this paper, we modeled the saturated elongation process through a Michaelis-Menten like mechanism, which is constituted by two sub-steps -- unspecific association and dissociation of a monomer with the fibril end, and subsequent conformational change of the associated monomer to fit itself to the fibrillar structure. Typical saturation concentrations were found to be $7-70mu M$ for A$beta$40, $alpha$-synuclein and etc. Furthermore, by using a novel Hamiltonian formulation, analytical solutions valid for both weak and strong saturated conditions were constructed and applied to the fibrillation kinetics of $alpha$-synuclein and silk fibroin.

Phase Transition Dynamics and Stochastic Resonance in Topologically Confined Nematic Liquid Crystals

Topological defects resulted from boundary constraints in confined liquid crystals have attracted extensive research interests. In this paper, we use numerical simulation to study the phase transition dynamics in the context of stochastic resonance in a bistable liquid crystal device containing defects. This device is made of nematic liquid crystals confined in a shallow square well, and is described by the planar Lebwohl-Lasher model. The stochastic phase transition processes of the system in the presence of a weak oscillating potential is simulated using an over-damped Langevin dynamics. Our simulation results reveal that, depending on system size, the phase transition may follow two distinct pathways: in small systems the pre-existing defect structures at the corners hold until the last stage and there is no newly formed defect point in the bulk during the phase transition, In large systems new defect points appear spontaneously in the bulk and eventually merge with the pre-existing defects at the corners. For both transition pathways stochastic resonance can be observed, but show dramatic difference in their responses to the boundary anchoring strength. In small systems we observe a sticky-boundary effect for a certain range of anchoring strength in which the phase transition gets stuck and stochastic resonance becomes de-activated. Our work demonstrates the dynamical interplay among defects, noises, and boundary conditions in confined liquid crystals.

The Markov Process Admits a Consistent Steady-State Thermodynamic Formalism

The seek for a new universal formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that: 1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equation fully agrees with that for the master equation; 2) a similar one-to-one correspondence could be established rigorously between the master equation and Fokker-Planck equation in the limit of large system size; 3) when a Markov process is restrained to one-step jump, the steady-state thermodynamic formalism for the Fokker-Planck equation with discrete state variables also goes to that for master equations, as the discretization step gets smaller and smaller. Our analysis indicated that, with respect to the steady state, general Markov processes admit a unified and self-consistent non-equilibrium thermodynamic formulation, regardless of underlying detailed models.

Thermodynamics of Markov Processes with Non-extensive Entropy and Free Energy

Statistical thermodynamics of small systems shows dramatic differences from normal systems. Parallel to the recently presented steady-state thermodynamic formalism for master equation and Fokker-Planck equation, we show that a ``thermodynamic theory can also be developed based on Tsallis generalized entropy $S^{(q)}=sum_{i=1}^N(p_i-p_i^q)/[q(q-1)]$ and Shiinos generalized free energy $F^{(q)}=[sum_{i=1}^Np_i(p_i/pi_i)^{q-1}-1]/[q(q-1)]$, where $pi_i$ is the stationary distribution. $dF^{(q)}/dt=-f_d^{(q)}le 0$ and it is zero iff the system is in its stationary state. $dS^{(q)}/dt-Q_{ex}^{(q)} = f_d^{(q)}$ where $Q_{ex}^{(q)}$ characterizes the heat exchange. For systems approaching equilibrium with detailed balance, $f_d^{(q)}$ is the product of Onsagers thermodynamic flux and force. However, it is discovered that the Onsagers force is non-local. This is a consequence of the particular transformation invariance for zero energy of Tsallis statistics.