It is known that polystyrene melts behave anomalously in fast elongational flows insofar as the steady-state elongational viscosity keeps decreasing with increasing stretching rate ε, without showing the typical upturn at ετ R ≈ 1, with τ R the Rouse time. The authors have recently suggested that such an anomalous behavior might be due to a decrease of the monomeric friction coefficient brought about by alignment of the Kuhn segments of the polymer to the stretching direction. Here we perform a quantitative analysis of such a possibility by first determining from existing stress−optical data how such a reduction should correlate to the order parameter of the Kuhn segments and then by performing nonequilibrium molecular dynamics (NEMD) simulations over a sequence of styrene oligomers. We have used NEMD not only to obtain diffusion coefficients of those oligomers but also, for what seems to be the first time, friction coefficients. Indeed, it is well-known that the Einstein relationship linking friction to diffusion does not hold true far from equilibrium. The friction coefficients so obtained correlate to the order parameter of the monomers in much the same way as in the polymeric case, and by increasing the length (or mass) of the oligomer, they appear to approach a similar characteristic curve.

This article introduces the notion of Generalized Poisson-Kac (GPK) processes which generalize the class of "telegrapher's noise dynamics" introduced by Marc Kac [39] in 1974, using Poissonian stochastic perturbations. In GPK processes the stochastic perturbation acts as a switching amongst a set of stochastic velocity vectors controlled by a Markov-chain dynamics. GPK processes possess trajectory regularity (almost everywhere) and asymptotic Kac limit, namely the convergence towards Brownian motion (and to stochastic dynamics driven by Wiener perturbations), which characterizes also the long-term/longdistance properties of these processes. In this article we introduce the structural properties of GPK processes, leaving all the physical implications to part II and part III [80,81].In statistical physics of gases, liquids, polymers and colloids (soft-matter physics, for short), the use of Brownian motion and Wiener processes can be regarded as the legacy of a large-number ansatz, wherein the influence of a manifold of small contributions, uniquely characterized by their mean and variance justifies the application of stochastic perturbations characterized by uncorrelated increments distributed in a normal way, which is precisely the definition of a Wiener process [6,7,8].The statistical characterization of microdynamic equations written in the form of Wiener-Langevin stochastic dynamics leads (using any representation of the stochastic integrals in the meaning of Ito, Stratonovich, Klimontovich, etc.) to parabolic (forward Fokker-Planck) equations for the probability density function, in which a deterministic drift v(x), and a tensorial diffusivity D can be always identified, possessing the structure of a second-order advection-diffusion equation.The analogy between Fokker-Planck equations and advection-diffusion problems permits to identify the overall probability flux J p (x, t) associated with the probability density p(x, t) as Basic principlesThe structure of one-dimensional Poisson-Kac processes, reviewed in the Appendix, contains the germs for its generalization in higher dimensions. Three basic principles can be enucleated out of it, that can guide the development of a simple and consistent stochastic theory of undulatory transport phenomena. These principles are:• The principle of stochastic reality;• The principle of the primitive variables;• The principle of the asymptotic Kac convergence.Below, their meaning and importance is outlined and discussed. 7

Nonlinear Elongational Rheology of Unentangled Polystyrene and Poly(p-Tert-Butylstyrene) Melts. Macromolecules 2018, 51 (23), 9710−9729] are here compared to the predictions of Brownian simulations of Fraenkel chains endowed with a monomeric friction coefficient that decreases with increasing the order parameter of the Kuhn segments. The model compares favorably with all startup data if a time delay in reducing the monomeric friction coefficient is accounted for in the fastest flows, a feature already observed by Matsumiya et al. The same model also describes the relaxation data except for a minor discrepancy in the longest relaxation time, probably due to the imperfect monodispersity of the sample and/or to a minor influence of topological interactions, not strong enough to be called entanglements. A discrepancy is also observed at the beginning of relaxation after fast flows, the origin of which remains unexplained.

This third part extends the theory of Generalized Poisson-Kac (GPK) processes to nonlinear stochastic models and to a continuum of states. Nonlinearity is treated in two ways: (i) as a dependence of the parameters (intensity of the stochastic velocity, transition rates) of the stochastic perturbation on the state variable, similarly to the case of nonlinear Langevin equations, and (ii) as the dependence of the stochastic microdynamic equations of motion on the statistical description of the process itself (nonlinear Fokker-Planck-Kac models). Several numerical and physical examples illustrate the theory. Gathering nonlinearity and a continuum of states, GPK theory provides a stochastic derivation of the nonlinear Boltzmann equation, furnishing a positive answer to the Kac's program in kinetic theory. The transition from stochastic microdynamics to transport theory within the framework of the GPK paradigm is also addressed.

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In this second part, we analyze the dissipation properties of Generalized Poisson-Kac (GPK) processes, considering the decay of suitable L 2 -norms and the definition of entropy functions. In both cases, consistent energy dissipation and entropy functions depend on the whole system of primitive statistical variables, the partial probability density functions {p α (x, t)} N α=1 , while the corresponding energy dissipation and entropy functions based on the overall probability density p(x, t) do not satisfy monotonicity requirements as a function of time. Examples from chaotic advection (standard map coupled to stochastic GPK processes) illustrate this phenomenon. Some complementary physical issues are also addressed: the ergodicity breaking in the presence of attractive potentials, and the use of GPK perturbations to mollify stochastic field equations.

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Novel coarse-grained models for molecular dynamics of tridecanoin melts are here proposed as results of a coarse-graining step procedure. The procedure is implemented to develop three coarse-grained models of increasing number of particle types from two to four. Force fields are computed by minimization of the deviations of appropriate distribution functions of the coarse-grained models from those of a reference atomistic one. Density, diffusivity and shear viscosity are computed by numerical simulation and compared with experimental values. The ability of each model to describe liquid-solid transitions is also analyzed. In particular, the model with four types of coarse-grained beads shows a transition from a liquid to a crystal phase.