The Jarzynski identity can be applied to instances when a microscopic system is pulled repeatedly but quickly along some coordinate, allowing the calculation of an equilibrium free energy profile along the pulling coordinate from a set of independent
non-equilibrium trajectories. Using the formalism of Wiener stochastic path integrals in which we assign temperature-dependent weights to Langevin trajectories, we derive exact formulae for the temperature derivatives of the free energy profile. This leads naturally to analytical expressions for decomposing a free energy profile into equilibrium entropy and internal energy profiles from non-equilibrium pulling. This decomposition can be done from trajectories evolved at a unique temperature without repeating the measurement as done in finite-difference decompositions. Three distinct analytical expressions for the entropy-energy decomposition are derived: using a time-dependent generalization of the weighted histogram analysis method, a quasi harmonic spring limit, and a Feynman-Kac formula. The three novel formulae of reconstructing the pair of entropy-energy profiles are exemplified by Langevin simulations of a two-dimensional model system prototypical for force-induced biomolecular conformational changes. Connections to single-molecule experimental means to probe the functionals needed in the decomposition are suggested.
Dynamics of a coarse-grained model for the room-temperature ionic liquid, 1-ethyl-3-methylimidazolium hexafluorophosphate, couched in the united-atom site representation are studied via molecular dynamics simulations. The dynamically heterogeneous be
havior of the model resembles that of fragile supercooled liquids. At or close to room temperature, the model ionic liquid exhibits slow dynamics, characterized by nonexponential structural relaxation and subdiffusive behavior. The structural relaxation time, closely related to the viscosity, shows a super-Arrhenius behavior. Local excitations, defined as displacement of an ion exceeding a threshold distance, are found to be mainly responsible for structural relaxation in the alternating structure of cations and anions. As the temperature is lowered, excitations become progressively more correlated. This results in the decoupling of exchange and persistence times, reflecting a violation of the Stokes-Einstein relation.
To understand the non-exponential relaxation associated with solvation dynamics in the ionic liquid 1-ethyl-3-methylimidazolium hexafluorophosphate, we study power spectra of the fluctuating Franck-Condon energy gap of a diatomic probe solute via mol
ecular dynamics simulations. Results show 1/f dependence in a wide frequency range over 2 to 3 decades, indicating distributed relaxation times. We analyze the memory function and solvation time in the framework of the generalized Langevin equation using a simple model description for the power spectrum. It is found that the crossover frequency toward the white noise plateau is directly related to the time scale for the memory function and thus the solvation time. Specifically, the low crossover frequency observed in the ionic liquid leads to a slowly-decaying tail in its memory function and long solvation time. By contrast, acetonitrile characterized by a high crossover frequency and (near) absence of 1/f behavior in its power spectra shows fast relaxation of the memory function and single-exponential decay of solvation dynamics in the long-time regime.
We construct equilibrium networks by introducing an energy function depending on the degree of each node as well as the product of neighboring degrees. With this topological energy function, networks constitute a canonical ensemble, which follows the
Boltzmann distribution for given temperature. It is observed that the system undergoes a topological phase transition from a random network to a star or a fully-connected network as the temperature is lowered. Both mean-field analysis and numerical simulations reveal strong first-order phase transitions at temperatures which decrease logarithmically with the system size. Quantitative discrepancies of the simulation results from the mean-field prediction are discussed in view of the strong first-order nature.
We consider the Ising model on a small-world network, where the long-range interaction strength $J_2$ is in general different from the local interaction strength $J_1$, and examine its relaxation behaviors as well as phase transitions. As $J_2/J_1$ i
s raised from zero, the critical temperature also increases, manifesting contributions of long-range interactions to ordering. However, it becomes saturated eventually at large values of $J_2/J_1$ and the system is found to display very slow relaxation, revealing that ordering dynamics is inhibited rather than facilitated by strong long-range interactions. To circumvent this problem, we propose a modified updating algorithm in Monte Carlo simulations, assisting the system to reach equilibrium quickly.
We study the collective behavior of an Ising system on a small-world network with the interaction $J(r) propto r^{-alpha}$, where $r$ represents the Euclidean distance between two nodes. In the case of $alpha = 0$ corresponding to the uniform interac
tion, the system is known to possess a phase transition of the mean-field nature, while the system with the short-range interaction $(alphatoinfty)$ does not exhibit long-range order at any finite temperature. Monte Carlo simulations are performed at various values of $alpha$, and the critical value $alpha_c$ beyond which the long-range order does not emerge is estimated to be zero. Thus concluded is the absence of a phase transition in the system with the algebraically decaying interaction $r^{-alpha}$ for any nonzero positive value of $alpha$.
