The Hydrodynamic Limit for Local Mean-Field Dynamics with Unbounded Spins

We consider the dynamics of a class of spin systems with unbounded spins interacting with local mean-field interactions. We prove convergence of the empirical measure to the solution of a McKean–Vlasov equation in the hydrodynamic limit and propagation of chaos. This extends earlier results of Gärtner, Comets and others for bounded spins or strict mean-field interactions.     

High Dimensional Atomic States of Hydrogenic Type: Heisenberg-like and Entropic Uncertainty Measures

High dimensional atomic states play a relevant role in a broad range of quantum fields, ranging from atomic and molecular physics to quantum technologies. The D-dimensional hydrogenic system (i.e., a negatively-charged particle moving around a positively charged core under a Coulomb-like potential) is the main prototype of the physics of multidimensional quantum systems. In this work, we review the leading terms of the Heisenberg-like (radial expectation values) and entropy-like (Rényi, Shannon) uncertainty measures of this system at the limit of high D. They are given in a simple compact way in terms of the space dimensionality, the Coulomb strength and the state’s hyperquantum numbers. The associated multidimensional position–momentum uncertainty relations are also revised and compared with those of other relevant systems.     

About the Definition of the Local Equilibrium Lattice Temperature in Suspended Monolayer Graphene

The definition of temperature in non-equilibrium situations is among the most controversial questions in thermodynamics and statistical physics. In this paper, by considering two numerical experiments simulating charge and phonon transport in graphene, two different definitions of local lattice temperature are investigated: one based on the properties of the phonon–phonon collision operator, and the other based on energy Lagrange multipliers. The results indicate that the first one can be interpreted as a measure of how fast the system is trying to approach the local equilibrium, while the second one as the local equilibrium lattice temperature. We also provide the explicit expression of the macroscopic entropy density for the system of phonons, by which we theoretically explain the approach of the system toward equilibrium and characterize the nature of the equilibria, in the spatially homogeneous case.     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes

We propose a novel framework to describe the time-evolution of dilute classical and quantum gases, initially out of equilibrium and with spatial inhomogeneities, towards equilibrium. Briefly, we divide the system into small cells and consider the local equilibrium hypothesis. We subsequently define a global functional that is the sum of cell H-functionals. Each cell functional recovers the corresponding Maxwell–Boltzmann, Fermi–Dirac, or Bose–Einstein distribution function, depending on the classical or quantum nature of the gas. The time-evolution of the system is described by the relationship dH/dt≤0, and the equality condition occurs if the system is in the equilibrium state. Via the variational method, proof of the previous relationship, which might be an extension of the H-theorem for inhomogeneous systems, is presented for both classical and quantum gases. Furthermore, the H-functionals are in agreement with the correspondence principle. We discuss how the H-functionals can be identified with the system’s entropy and analyze the relaxation processes of out-of-equilibrium systems.     

Entropy of classical systems with long-range interactions

We discuss the form of the entropy for classical Hamiltonian systems with long-range interaction using the Vlasov equation which describes the dynamics of a N particle in the limit N-->infinity. The stationary states of the Hamiltonian system are subject to infinite conserved quantities due to the Vlasov dynamics. We show that the stationary states correspond to an extremum of the Boltzmann-Gibbs entropy, and their stability is obtained from the condition that this extremum is a maximum. As a consequence, the entropy is a function of an infinite set of Lagrange multipliers that depend on the initial condition. We also discuss in this context the meaning of ensemble inequivalence and the temperature.     

Kullback–Leibler Divergence of a Freely Cooling Granular Gas

Finding the proper entropy-like Lyapunov functional associated with the inelastic Boltzmann equation for an isolated freely cooling granular gas is a still unsolved challenge. The original H-theorem hypotheses do not fit here and the H-functional presents some additional measure problems that are solved by the Kullback–Leibler divergence (KLD) of a reference velocity distribution function from the actual distribution. The right choice of the reference distribution in the KLD is crucial for the latter to qualify or not as a Lyapunov functional, the asymptotic “homogeneous cooling state” (HCS) distribution being a potential candidate. Due to the lack of a formal proof far from the quasielastic limit, the aim of this work is to support this conjecture aided by molecular dynamics simulations of inelastic hard disks and spheres in a wide range of values for the coefficient of restitution (α) and for different initial conditions. Our results reject the Maxwellian distribution as a possible reference, whereas they reinforce the HCS one. Moreover, the KLD is used to measure the amount of information lost on using the former rather than the latter, revealing a non-monotonic dependence with α.     

Lyapunov exponents in unstable systems

We investigate the dynamical behavior of unstable systems in the vicinity of the critical point associated with a liquid-gas phase transition. By considering a mean-field treatment, we first perform a linear analysis and discuss the instability growth times. Then, coming to complete Vlasov simulations, we investigate the role of nonlinear effects and calculate the Lyapunov exponents. As a main result, we find that near the critical point, the Lyapunov exponents exhibit a power-law behavior, with a critical exponent beta=0.5. This suggests that in thermodynamical systems the Lyapunov exponent behaves as an order parameter to signal the transition from the liquid to the gas phase.     

Vlasov equation on a symplectic leaf

The infinite dimensional phase space of the Vlasov equation is foliated by symplectic manifolds (leaves) which are invariant under the dynamics. By adopting a Lie transform representation, exp{W, }, for near-identity canonical transformations we obtain a local coordinate system on a leaf. The evolution equation defined by restricting the Vlasov equation to the leaf is approximately represented by the evolution of W. We derive the equation for ∂_tW and show that it is hamiltonian relative to the nondegenerate Kirillov-Kostant-Souriau symplectic structure.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.     

Transport Properties for a Collisionless Plasma from Finite-Larmor-Radius Equations in General Magnetic Fields

We have take the Vlasov equation in a local system of coordinates defined by the curved magnetic field and we have expanded the distribution function in terms of a small parameter ? which is the ratio of ion Larmor radius to the characteristic length in the direction perpendicular to magnetic field. Assuming an approximation for the distribution function, which is consistent with physical systems, we have obtained transport properties in different orders of approximation taking account of the general heat flux vector. We find that there are additional terms appearing in the components of stresses due to the parallel energy flow term in the heat flux vector.     

Vlasov moments, integrable systems and singular solutions

The Vlasov equation governs the evolution of the single-particle probability distribution function (PDF) for a system of particles interacting without dissipation. Its singular solutions correspond to the individual particle motions. The operation of taking the moments of the Vlasov equation is a Poisson map. The resulting Lie-Poisson Hamiltonian dynamics of the Vlasov moments is found to be integrable is several cases. For example, the dynamics for coasting beams in particle accelerators is associated by a hodograph transformation to the known integrable Benney shallow-water equation. After setting the context, the Letter focuses on geodesic Vlasov moment equations. Continuum closures of these equations at two different orders are found to be integrable systems whose singular solutions characterize the geodesic motion of the individual particles.     

Quantum dynamics of Gaudin magnets

Quantum dynamics of many-body systems is a fascinating and significant subject for both theory and experiment. The question of how an isolated many-body system evolves to its steady state after a sudden perturbation or quench still remains challenging. In this paper, using the Bethe ansatz wave function, we study the quantum dynamics of an inhomogeneous Gaudin magnet. We derive explicit analytical expressions for various local dynamic quantities with an arbitrary number of flipped bath spins, such as: the spin distribution function, the spin–spin correlation function, and the Loschmidt echo. We also numerically study the relaxation behavior of these dynamic properties, gaining considerable insight into coherence and entanglement between the central spin and the bath. In particular, we find that the spin–spin correlations relax to their steady value via a nearly logarithmic scaling, whereas the Loschmidt echo shows an exponential relaxation to its steady value. Our results advance the understanding of relaxation dynamics and quantum correlations of long-range interacting models of the Gaudin type.     

Finite-space Lyapunov exponents and pseudochaos

We propose a definition of finite-space Lyapunov exponent. For discrete-time dynamical systems, it measures the local (between neighboring points) average spreading of the system. We justify our definition by showing that, for large classes of chaotic maps, the corresponding finite-space Lyapunov exponent approaches the Lyapunov exponent of a chaotic map when M-->infinity, where M is the cardinality of the discrete phase space. In analogy with continuous systems, we say the system has pseudochaos if its finite-space Lyapunov exponent tends to a positive number (or to +infinity), when M-->infinity.     

Thermal relaxation of systems with quadratic heat bath coupling: I. Classical systems

We consider the evolution of systems whose coupling to the heat bath is quadratic in the bath coordinates. Performing an explicit elimination of the bath variables we arrive at an equation of evolution for the system variables alone. In the weak coupling limit we show that the equation is of the generalized Langevin form, with fluctuations that are Gaussian and that obey a fluctuation-dissipation relation. If the system-bath coupling is linear in the system coordinates the resulting fluctuations are additive and the dissipation is linear. If the coupling is nonlinear in the system coordinates, the resulting fluctuations are multiplicative and the dissipation is nonlinear.     

The Information Loss of a Stochastic Map

We provide a stochastic extension of the Baez–Fritz–Leinster characterization of the Shannon information loss associated with a measure-preserving function. This recovers the conditional entropy and a closely related information-theoretic measure that we call conditional information loss. Although not functorial, these information measures are semi-functorial, a concept we introduce that is definable in any Markov category. We also introduce the notion of an entropic Bayes’ rule for information measures, and we provide a characterization of conditional entropy in terms of this rule.     

Heterologous production of fungal natural products: Reconstitution of biosynthetic gene clusters in model host Aspergillus oryzae

While exploring phytotoxic metabolites from phytopathogenic fungi in the 1970s, we became interested in biosynthetic enzymes that catalyze Diels–Alder reactions involving biosynthesis of several phytotoxins that we isolated. Target enzymes were successfully characterized, and this triggered the identification of various Diels–Alderases in a recent decade. Through our Diels–Alderase project in 1990s, we recognized a highly efficient expression system of various biosynthetic genes with Aspergillus oryzae as a host. With the development of tools such as genomic data and bioinformatics analysis to identify biosynthetic gene clusters for natural products, we developed a highly reliable methodology such as hot spot knock-in to elucidate the biosynthetic pathways of representative fungal metabolites including phytotoxic substances. This methodology allows total biosynthesis of natural products and genome mining using silent biosynthetic gene clusters to obtain novel bioactive metabolites. Further applications of this technology are discussed.     

Holonomy for gerbes over orbifolds

In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan’s theory of twisted orbifold cohomology [1] and the local system of Freed–Hopkins–Teleman in their work in twisted K-theory [2]. In the case in which the orbifold is simply a manifold we recover previous results of Gawȩdzki [3] and Brylinski[4].     

Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator

With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the different terms in a stochastic differential equation. With the full representation of this operator, we are able to separate finite-time corrections of the power-series expansion of arbitrary order into terms with and without derivatives of the Kramers–Moyal coefficients. We arrive at a closed-form solution expressed through conditional moments, which can be extracted directly from time-series data with a finite sampling intervals. We provide all finite-time correction terms for parametric and non-parametric estimation of the Kramers–Moyal coefficients for discontinuous processes which can be easily implemented—employing Bell polynomials—in time-series analyses of stochastic processes. With exemplary cases of insufficiently sampled diffusion and jump-diffusion processes, we demonstrate the advantages of our arbitrary-order finite-time corrections and their impact in distinguishing diffusion and jump-diffusion processes strictly from time-series data.