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The kinetics and thermodynamics of free living copolymerization are studied for processes with rates depending on k monomeric units of the macromolecular chain behind the unit that is attached or detached. In this case, the sequence of monomeric units in the growing copolymer is a kth-order Markov chain. In the regime of steady growth, the statistical properties of the sequence are determined analytically in terms of the attachment and detachment rates. In this way, the mean growth velocity as well as the thermodynamic entropy production and the sequence disorder can be calculated systematically. These different properties are also investigated in the regime of depolymerization where the macromolecular chain is dissolved by the surrounding solution. In this regime, the entropy production is shown to satisfy Landauer's principle.

The thermodynamics of copolymerization processes leading to the formation and replication of copolymers is presented. Copolymers are natural one-dimensional supports of information, which may be coded in the sequence of monomeric units randomly attached or detached during their synthesis. Multivariate fluctuation relations are here established for copolymerization processes synthesizing Bernoulli and first-order Markov chains. Thereof, the thermodynamic entropy production is deduced and shown to depend on quantities characterizing information possibly stored in the copolymer sequence.

For a piecewise linear intermittent map, the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the Frobenius–Perron operator ^P are explicitly derived. The evolution of the averages are shown to be a superposition of the contributions from two simple eigenvalues 1 and λ d ∈(−1, 0), and a continuous spectrum on the unit interval [0,1] of ^P. Power-law decay of correlations are controlled by the continuous spectrum. Also the non-normalizable invariant measure in the non-stationary regime is shown to determine the strength of the power-law decay.

Dynamical instability is studied in a deterministic dynamical system of Hamiltonian type composed of a tracer particle in a fluid of many particles. The tracer and fluid particles are hard balls (disks, in two dimensions, or spheres, in three dimensions) undergoing elastic collisions. The dynamical instability is characterized by the spectrum of Lyapunov exponents. The tracer particle is shown to dominate the Lyapunov spectrum in the neighborhoods of two limiting cases: the Lorentz-gas limit in which the tracer particle is much lighter than the fluid particles and the Rayleigh-flight limit in which the fluid particles have a vanishing radius and form an ideal gas. In both limits, a gap appears in the Lyapunov spectrum between the few largest Lyapunov exponents associated with the tracer and the rest of the Lyapunov spectrum.

A trace formula is derived for the Fokker–Planck equation associated with Ito stochastic differential equations describing noisy time-continuous nonlinear dynamical systems. In the weak-noise limit, the trace formula provides estimations of the eigenvalues of the Fokker–Planck operator on the basis of the Pollicott–Ruelle resonances of the noiseless deterministic system, which is assumed to be non-bifurcating. At first order in the noise amplitude, the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo-energy of the Onsager–Machlup–Freidlin–Wentzell scheme.

For a previously introduced conservative multibaker map with energy, the Gaspard–Gilbert–Dorfman entropy production of the stationary state induced by the flux boundary condition is calculated and the entropy production is shown (i) to be nonnegative, (ii) to vanish in the fine-grained limit for finite chains, (iii) to take the phenomenologically expected value in the middle of the chain and to deviate from it near the boundaries, and (iv) to reduce to the phenomenological expression in the scaling limit where the lattice site n∈Z and time t∈Z are scaled respectively as n=LξX and t=LτT and the limits of Lξ→+∞ and Lτ→+∞ are taken while keeping the diffusion coefficient D=lLτ/L2ξ constant, l being the transition rate of the model. The mass and heat transports are also studied in the scaling limit under an additional assumption that the edges of the chain are in equilibrium with different temperatures. In the linear heat transport regime, Fourier's law of heat conduction and the thermodynamic expression of the associated entropy production are obtained.

We derive a formula for the level spacing probability distribution in quantum graphs. We apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases. Moreover, we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs.

We apply the hypothesis of microscopic chaos to diffusion-controlled reaction which we study in a reactive periodic Lorentz gas. The relaxation rate of the reactive eigenmodes is obtained as eigenvalue of the Frobenius–Perron operator, which determines the reaction rate. The cumulative functions of the eigenstates of the Frobenius–Perron operator are shown to be generalizations of Lebesgue's singular continuous functions. For small enough densities of catalysts, the reaction is controlled by the diffusion. A random-walk model of this diffusion-controlled reaction process is presented, which is used to study the dependence of the reaction rate on the density of catalysts.

A multibaker map with “kinetic energy” is proposed which incorporates an external field. The map is volume-preserving, time-reversal symmetric and conserves total energy. In an appropriate macroscopic limit, the particle distribution is shown to obey a Smoluchowski-type equation. For the cases without any external field and with a constant external force, the nonequilibrium stationary states are constructed by solving the evolution equation of the partially integrated distribution functions. These states are described by singular functions such as incomplete Takagi functions and Lebesgue's singular functions. In an appropriate macroscopic limit, the mass flows for the stationary states are shown to be identical to the ones expected from the Smoluchowski equation and a “heat flow” proportional to the local energy gradient appears. The Gaspard–Gilbert–Dorfman entropy production is calculated for the stationary states and is shown to be positive. Particularly, for the case with a constant external force, when the energy distribution is independent of the spatial distribution, the entropy production reduces to the one consistent with classical thermodynamics. The result shows that there exists a volume-preserving driven multibaker map whose entropy production is consistent with classical thermodynamics.

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diffusion process like the multibaker map or the Lorentz gas. A continuous flux of particles is imposed across the system resulting in a steady gradient of concentration. In the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondifferentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be defined as a coarse-grained entropy instead of the fined-grained Gibbs entropy, which would require the existence of a regular measure with a density. The coarse-grained entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics.

We present a treatment of many-body Fermionic systems that facilitates an expression of the well-known quantities in a series expansion of the Planck's constant. The ensuing semiclassical result contains to a leading order of the response function the classical time correlation function of the observable followed by the Weyl-Wigner series, on top of these terms are the periodic-orbit correction terms. The treatment given here starts from linear response assumption of the many-body theory and in its connection with semiclassical theory, it makes no assumption of the integrability of classical dynamics underlying the one-body quantal system. Applications of the framework are also discussed., Comment: 18 pages, TeX

Nonequilibrium stationary states are studied for a multibaker map, a simple reversible chaotic dynamical system. The probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function. The equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of deRham-type functional equations. To select the physical states, the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type. This system corresponds to a diffusive flow experiment through a slab of material. For long times, any initial distribution approaches the stationary one obeying Fick's law. At stationarity, the intracell distribution is singular in the stable direction and expressed by the Takagi function, which is continuous but has no finite derivatives. The result suggests that singular measures play an important role in the dynamical description of non-equilibrium states.

We study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area-preserving mapping of an infinitely extended strip onto itself. This exactly solvable mapping sustains chaotic behaviors and diffusion processes. The relationship between the diffusion coefficient, the Lyapunov exponent, and the entropy per unit time is derived. The long-lived classical resonances of the Liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation. In this sense, there is a quasi-isomorphism between the resonance spectrum of the Liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation. Furthermore, we show that a fractal repeller is associated to each non-equilibrium state in the isolated and finite multibaker chain. The nonequilibrium states are all unstable with respect to the equilibrium, validating a weak form of the second principle of thermodynamics for the present dynamical system. Consequences of nonequilibrium fractals on classical measurements are discussed. We then describe the open multibaker chain as a scattering system. Fractal properties of chaotic scattering are here shown to be related to diffusion in the chain.

In this paper, we study the structural properties of liquid group-V elements (and a compound InSb) and we analyse the differences with their room temperature crystalline phases. Neutron diffraction experiments are performed using short-wavelength neutrons (0.7 A). We show different melting behaviours of a light element (As) and a heavy element (Sb) under study. Whereas As keeps its coordination three upon melting, Sb is six-coordinated in the melt. Similarly, InSb has a coordination number increasing from 4 to about 6 upon melting. Large entropies of melting are related to important structural variations.

We show that a simple tight binding model with a repulsive potential describes the Peierls distortions in covalent systems and the well-known octet rule. The existence and the intensity of the Peierls distortion is mainly related to the hardness of the repulsive potential as demonstrated both by theoretical calculations and by the experimental systematic analysis of liquid structures. In particular, As is threefold coordinated and Sb is sixfold coordinated in the liquid, the qualitative difference is explained by the ratio of the distortion energy ΔE to the thermal energy kBT. The Asx Sb1-x alloys show continuously varying average coordination numbers showing: the semiconductor-metal transition is continuous with concentration.In addition, we illustrate in the case of liquid Se that, tight binding Monte Carlo simulations are able to describe quantitatively the structure of liquid elements provided the Van der Waals potential is added.

A concept of time-reversed entropy per unit time is introduced in analogy with the entropy per unit time by Shannon, Kolmogorov, and Sinai. This time-reversed entropy per unit time characterizes the dynamical randomness of a stochastic process backward in time, while the standard entropy per unit time characterizes the dynamical randomness forward in time. The difference between the time-reversed and standard entropies per unit time is shown to give the entropy production of Markovian processes in nonequilibrium steady states.

We presented experimental evidence for the existence of microscopic chaos in the mesoscopic motion of a brownian particle in solution. We used standard techniques to analyse long trajectories of a brownian particle and inferred a dynamical entropy from this analysis. We showed that this dynamical entropy can be accessed experimentally by measuring multiple-time correlation functions for the particle. The dynamical entropy was then used to provide a positive lower bound for the sum of the Lyapunov exponents for the underlying deterministic dynamical system composed of the fluid and brownian particles. We concluded that the positive dynamical entropy, obtained experimen-tally, was evidence for the existence of positive Lyapunov exponents in the underlying dynamics, and hence for the existence of microscopic chaos generated by a dynamical instability.

State diagrams of two model systems involving three variables are constructed. The parameter dependence of different forms of complex nonperiodic behavior, and particularly of homoclinic orbits, is analyzed. It is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system.

The bifurcations of periodic orbits in a class of autonomous three-variable, nonlinear-differential-equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues (ρ ±iω,λ), where ¦ρ/λ¦ < 1 (Sil'nikov's condition), are studied in a two-parameter space. The perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period-doubling bifurcations leading to periodic orbits which may be attractors if ¦ρ/λ¦ < 1/2. The accumulation rate of the critical parameter values at the homoclinic system is exp(-2π¦ρ/ω¦). A global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed. Cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity. A countable set of these cusp bifurcations with scaling properties related to the eigenvaluesρ±iω of the stationary state are shown to occur in infinitely contractive flows. In the two-parameter space, the periodic orbit attractor domain exhibits a spiral structure globally, around the set of homoclinic systems, in which all the different periodic orbits are continuously connected.

Nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated. We study the bifurcation sequences of the mixed-mode oscillations generated by the homoclinicity, which are shown to belong to two different types, depending on the nature of the Liapunov numbers of the basic periodic orbit. A detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation. Chaotic cool flame oscillations are also observed in the same model. When the control parameter crosses a critical value, this chaotic set of trajectories becomes globally unstable and forms a Cantor-like hyperbolic repellor, and the ignition mechanism generates ahomoclinic tangency to the Cantor set of trajectories. The complex bifurcation diagram may be globally reconstructed from a one-dimensional dynamical system, thanks to the strong contractivity of thermokinetics. It is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system.