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Intermolecular correlations lower values of both diffusion and entropy. We present an analysis of the existing relations between long-time diffusion (D) and entropy. S. A recently proposed inequality, a lower bound, by Sorkin et al., expresses the long-time diffusion in terms of diffusion in a reference state and the entropy difference. Such a relationship may provide a measure of intermolecular correlations. We show that for a one-dimensional rugged energy landscape, the lower bound becomes equality only if certain three-site correlations are neglected. When these correlations are included, we can derive an accurate expression that agrees with computer simulations. The strong dimensionality dependence of diffusion of a Brownian particle in a rugged energy landscape also resembles the recently proposed inequality. We show that for interacting colloids, a mode-coupling theory-type calculation of diffusion coefficient can be combined with the new inequality to estimate entropy change from long-range inter-colloid correlations. Interestingly, the rate of barrier crossing in a multidimensional potential energy surface, related to diffusion in a periodic lattice, is shown to admit a relation that depends on the entropy change in a way reminiscent of the relation between D and S.

The celebrated Stokes Law (SL) of hydrodynamics predicts that the velocity of a particle pulled through a liquid by an external force, Fex, is directly proportional to the force and inversely proportional to the friction {\zeta} acted by the medium on the particle. We investigate the range of validity of Stokes Law at molecular length scales by employing computer simulations to calculate friction by pulling a tagged particle with a constant force. We thus calculate friction for two model interaction potentials, Lennard-Jones, and soft sphere, for several particle sizes, ranging from radius (a) smaller than the solvent particles to three times larger. We next obtain friction from diffusion (D) by using Einstein's relation between diffusion and friction {\zeta} in an unperturbed liquid. We find, to our surprise, a quantitative agreement between the two at a small-to-intermediate pulling force regime for all the sizes studied. The Law does break down at a large pulling force beyond a threshold value. Importantly, the range of validity of Stokes' scheme to obtain friction increases substantially if we turn off the attractive part of the interaction potential. Additionally, we calculate the viscosity ({\eta}) of the unperturbed liquid and find a good agreement with the Stokes-Einstein relation {\zeta}=C{\eta}a for the viscosity dependence with a value of C close to 5 {\pi}, that is intermediate between the slip and stick boundary condition.

The task of a first principles theoretical calculation of the rate of gas to liquid nucleation has remained largely incomplete despite the existence of reliable results from unbiased simulation studies at large supersaturation. Although the classical nucleation theory, formulated by Becker-Doring-Zeldovich (BDZ) about a century ago, provides an elegant, widely-used picture of nucleation in a first-order phase transition, the theory finds difficulties in predicting the rate accurately, especially in the case of gas-to-liquid nucleation. Here, we use a multiple-order parameter description to construct the nucleation free energy surface needed to calculate the nucleation rate. A multidimensional non-Markovian rate theory formulation that generalizes Langer's well-known nucleation theory by using Grote-Hynes multidimensional non-Markovian treatment is used to obtain the rate of barrier crossing. We find good agreement of the theory with the rate obtained by direct unbiased molecular dynamics simulations, the latter is feasible at large supersaturation, S. The theory gives the experimentally strong dependence of the rate of nucleation on supersaturation, S. Interestingly, we find a strong influence of frequency-dependent friction at the barrier top. This arises from multiple recrossing of the barrier surface. We find that a Markovian theory, such as Langer's formulation, fails to capture the rate quantitatively. In addition, the multidimensional transition state theory expression performs poorly, revealing the underlying role of friction.

Boltzmann's H-Theorem, formulated 150 years ago in terms of H-function that also bears his name, is one of the most celebrated theorems of science and paved the way for the development of nonequilibrium statistical mechanics. Nevertheless, quantitative studies of the H-function, denoted by H(t), in realistic systems are relatively scarce because of the difficulty of obtaining the time-dependent momentum distribution analytically. Also, the earlier attempts proceeded through the solution of Boltzmann's kinetic equation, which was hard. Here we investigate, by direct molecular dynamics simulations and analytic theory, the time dependence of H(t). We probe the sensitivity of nonequilibrium relaxation to interaction potential and dimensionality by using the H-function H(t). We evaluate H(t) for three different potentials in all three dimensions and find that it exhibits surprisingly strong sensitivity to these factors. The relaxation of H(t) is long in 1D, but short in 3D. We obtain, for the first time, a closed-form analytic expression for H(t) using the solution of the Fokker-Planck equation for the velocity space probability distribution and compare its predictions with the simulation results. Interestingly, H(t) is found to exhibit linear response when vastly different initial nonequilibrium conditions are employed. The oft-quoted relation of H-function with Clausius's entropy theorem is discussed.

Diffusion in a multidimensional energy surface with minima and barriers is a problem of importance in statistical mechanics and also has wide applications, such as protein folding. To understand it in such a system, we carry out theory and simulations of a tagged particle moving on a two-dimensional periodic potential energy surface, both in the presence and absence of noise. Langevin dynamics simulations at multiple temperatures are carried out to obtain the diffusion coefficient of a solute particle. Friction is varied from zero to large values. Diffusive motion emerges in the limit of long times, even in the absence of noise, although the trajectory is found to remain correlated over a long time. This correlation is manifested in correlated returns to the starting minima following a scattering by surrounding maxima. Noise destroys this correlation, induces chaos, and increases diffusion at small friction. Diffusion thus exhibits a non-monotonic friction dependence at the intermediate value of the damping, ultimately converging to our theoretically predicted value. The latter is obtained by using the well-established relation between diffusion and random walk. An excellent agreement is obtained between theory and simulations in the high friction limit, but not so in the intermediate regime. The rate of escape from one cell to another is obtained from the multidimensional rate theory of Langer. We find that enhanced dimensionality plays an important role. In order to quantify the effects of noise on the potential-imposed coherence on the trajectories, we calculate the Lyapunov exponent. At small friction values, the Lyapunov exponent mimics the friction dependence of the rate.

Although an intimate relation between entropy and diffusion has been advocated for many years and even seems to have been verified in theory and experiments, a quantitatively reliable study, and any derivation of an algebraic relation between the two does not seem to exist. Here we explore the nature of this entropy-diffusion relation in three deterministic systems where an accurate estimate of both can be carried out. We study three deterministic model systems, (a) the motion of a single point particle with constant energy in a two-dimensional periodic potential energy landscape, (b) the same in regular Lorentz gas where a point particle with constant energy moves between collisions with hard disc scatterers and (c) motion of a point particle among the boxes with small apertures. These models, introduced by Zwanzig, exhibit diffusive motion in the limit where ergodicity is shown to exist. We then explore the diffusion-entropy relation by an accurate calculation of both diffusion and entropy for the aforementioned model systems. We estimate the self-diffusion coefficient of the particle by employing computer simulations and entropy by quadrature using Boltzmann's formula. We observe an interesting crossover in the diffusion-entropy relation in some specific regions which is attributed to the emergence of correlated returns. The crossover could herald a breakdown of the Rosenfeld-like exponential scaling between the two, as observed at low temperatures. Later, we modify the scaling relation to account for the correlated motions and present a detailed analysis of the dynamical entropy obtained via Lyapunov exponent which is rather an important quantity in the study of deterministic systems., Comment: 29 pages, 8 figures, 1 table

Spherical confinement can alter the properties of a dipolar fluid in several different ways. In an atomistic molecular dynamics simulation study of two different dipolar liquids (SPC/E water and a model Stockmayer fluid) confined to nanocavities of different radii ranging from Rc=1nm to 4nm, we find that the Kirkwood correlation factor remains surprisingly small in water, but not so in model Stockmayer liquid. This gives rise to an anomalous ultrafast relaxation of the total dipole moment time correlation function (DMTCF). The static dielectric constant of water under nanoconfinement (computed by employing Clausius-Mossotti equation, the only exact relation) exhibits a strong dependence on the size of the nanocavity with a remarkably slow convergence to the bulk value. Interestingly, the value of the volume becomes ambiguous in this nanoworld. It is determined by the liquid-surface interaction potential and is to be treated with care because of the sensitivity of the Clausius-Mossotti equation to the volume of the nanosphere. We discover that the DMTCF for confined water exhibit a bimodal 1/f noise power spectrum. We also comment on the applicability of certain theoretical formalisms that become dubious in the nanoworld., Comment: 19 pages, 13 figures, 2 tables

Experiments reveal that DNA solvation dynamics (SD) is characterized by multiple time scales ranging from a few ps to hundreds of ns and in some cases even up to microseconds. The last part of decay is slow and is characterized by a power law (PL). The microscopic origin of this PL is not yet clearly understood. Here we present a theoretical study based on time dependent statistical mechanics and computer simulations. Our investigations show that the primary candidates responsible for this exotic nature of SD are the counterions and ions from the buffer solution. We employ the model developed by Oosawa for polyelectrolyte solution that includes effects of counterion fluctuations to construct a frequency dependent dielectric function. We use it in the continuum model of Bagchi, Fleming and Oxtoby only to find that it fails to explain the slow PL decay of DNA solvation dynamics. We then extend the model by employing the continuous time random walk technique developed by Scher-Montroll-Lax. This approach can explain the long time PL decay, in terms of the collective response of the counter ions. From MD simulations we find frequent occurrence of random walk of tagged counter ions along the phosphate backbone. We propose a generalized random walk model for counterion hopping and carry out kinetic Monte Carlo simulations to show that the nonexponential contribution to solvation dynamics can indeed arise from dynamics of such ions. We also employ a Mode Coupling Theory analysis to understand the slow relaxation that originates from ions in solution. Explicit evaluation suggests that buffer ion contribution could explain logarithmic time dependence in the ns time scale, but not a power law. From MD simulations we find log-normal distributions of relaxation times of water dynamics inside the grooves. This is responsible for the initial faster multiexponential decay of SD., Comment: 60 pages, 15 figures, 6 tables