Glauber dynamics of fluctuations

We derive the time evolution of the normal fluctuations of a classical lattice spin system induced by a generalized Glauber dynamics. The canonical form of this dynamics is derived. We prove that it is asymptotically (i.e., after the central limit) free. The results are applied to give a rigorous proof of the macroscopic reciprocity relations and the linear theory for small deviations from equilibrium.     

正则系综的变电荷分子动力学方法

在迭代变电荷方法的基础上加以改进得到适于正则系综的变电荷方法.利用正则系综的热浴方法补偿模拟过程中动能的衰减.分子动力学模拟的结果表明,改进的变电荷方法能够避免能量漂移问题,在相同的电荷精度条件下,所需的迭代次数减少,可提高计算效率.     

Wigner Rotations and Precession of Polarization

An easy and direct derivation of Thomas precession is obtained from infinitesimal Wigner rotations arising in unitary representations of the Poincaré group. For spin > 1/2, multipole parameters are studied from this point of view. The canonical 3-component definition of polarization arising naturally in this context is compared with formalisms which start form a pseudo 4-vector and an antisymmetric tensor respectively. The full Thomas equations, including Larmor precession, is derived using time derivatives of finite Wigner rotations. Exact solutions, with arbitrary initial conditions, are presented for constant magnetic fields and for orthogonal constant electric and magnetic fields. For a class of plane wave external fields exact solutions are obtained for the Dirac equation generalized by the inclusion of anomalous magnetic moment (Pauli) and electric dipole moment terms. Using the front form of dynamics, well-adapted to this context and coinciding with proper time dynamics, expectation values are calculated. The polarization pseudo 4-vector thus obtained is shown to satisfy the BMT equation, which is equivalent to the Thomas equation. This shows that the validity of the classical precession equations is not necessarily restricted to slowly varying external fields. These solutions can also be of interest in the study of spin 1/2 particles in laser fields and in the study of electric dipole moments.     

Anharmonic Oscillator Driven by Additive Ornstein–Uhlenbeck Noise

We present an analytical study of a nonlinear oscillator subject to an additive Ornstein–Uhlenbeck noise. Known results are mainly perturbative and are restricted to the large dissipation limit (obtained by neglecting the inertial term) or to a quasi-white noise (i.e., a noise with vanishingly small correlation time). Here, in contrast, we study the small dissipation case (we retain the inertial term) and consider a noise with finite correlation time. Our analysis is non perturbative and based on a recursive adiabatic elimination scheme a reduced effective Langevin dynamics for the slow action variable is obtained after averaging out the fast angular variable. In the conservative case, we show that the physical observables grow algebraically with time and calculate the associated anomalous scaling exponents and generalized diffusion constants. In the case of small dissipation, we derive an analytic expression of the stationary probability distribution function (PDF) which differs from the canonical Boltzmann–Gibbs distribution. Our results are in excellent agreement with numerical simulations.     

Classical and quantum q-deformed physical systems

On the basis of non-commutative q-calculus, we investigate a q-deformation of the classical Poisson bracket in order to formulate a generalized q-deformed dynamics in the classical regime. The obtained q-deformed Poisson bracket appears invariant under the action of the q-symplectic group of transformations. Within this framework we introduce the q-deformed Hamilton equations and we derive the evolution equation for some simple q-deformed mechanical systems governed by a scalar potential dependent only on the coordinate variable. It appears that the q-deformed Hamiltonian, which is the generator of the equation of motion, is generally not conserved in time but, in correspondence, a new constant of motion is generated. Finally, by following the standard canonical quantization rule, we compare the well-known q-deformed Heisenberg algebra with the algebra generated by the q-deformed Poisson bracket. PACS 02.45.Gh, 45.20.-d, 03.65.-w, 02.20.Uw     

Ambient space formulations and statistical mechanics of holonomically constrained Langevin systems

The most classic approach to the dynamics of an n-dimensional mechanical system constrained by d independent holonomic constraints is to pick explicitly a new set of (n − d) curvilinear coordinatesparametrizingthe manifold of configurations satisfying the constraints, and to compute the Lagrangian generating the unconstrained dynamics in these (n − d) configuration coordinates. Starting from this Lagrangian an unconstrained Hamiltonian H(q,p) on 2(n−d) dimensional phase space can then typically be defined in the standard way via a Legendre transform. Furthermore, if the system is in contact with a heat bath, the associated Langevin and Fokker-Planck equations can be introduced. Provided that an appropriate fluctuation-dissipation condition is satisfied, there will be a canonical equilibrium distribution of the Gibbs form exp(−βH) with respect to the flat measure dqdp in these 2(n − d) dimensional curvilinear phase space coordinates. The existence of (n − d) coordinates satisfying the constraints is often guaranteed locally by an implicit function theorem. Nevertheless in many examples these coordinates cannot be constructed in any tractable form, even locally, so that other approaches are of interest. In ambient space formulations the dynamics are defined in the full original n-dimensional configuration space, and associated 2n-dimensional phase space, with some version of Lagrange multipliers introduced so that the 2(n − d) dimensional sub-manifold of phase space implied by the holonomic constraints and their time derivative, is invariant under the dynamics. In this article we review ambient space formulations, and explain that for constrained dynamics there is in fact considerable freedom in how a Hamiltonian form of the dynamics can be constructed. We then discuss and contrast the Langevin and Fokker-Planck equations and their equilibrium distributions for the different forms of ambient space dynamics.     

On Relativistic Mass Spectra of a Two-Particle System

Abstract

A relativistic two-particle system with time-asymmetric scalar and vector interactions in the two-dimensional space-time is considered within the frame of the front form of dynamics using the dynamical symmetry approach. The mass-shell equation may be represented in terms of the nonlinear canonical realization of the Lie algebra of the group SO(2, 1). This allows us to quantize the system and to obtain a closed form for the mass spectrum.     

Nonlinear Waves in Nonuniform Complex Astrocloud

The global nonlinear gravito‐electrostatic eigen‐fluctuation behaviors in large‐scale non‐uniform complex astroclouds in quasi‐neutral hydrodynamic equilibrium are methodologically analyzed. Its composition includes warm lighter electrons, ions; and massive bi‐polar multi‐dust grains (inertial) with partial ionization sourced, via plasma‐contact electrification, in the cloud plasma background. The multi‐fluidic viscous drag effects are conjointly encompassed. The naturalistic equilibrium inhomogeneities, gradient forces and nonlinear convective dynamics are considered without any recourse to the Jeans swindle against the traditional perspective. An inho‐mogeneous multiscale analytical method is meticulously applied to derive a new conjugated non‐integrable coupled (via zeroth‐order factors) pair of variable‐coefficient inhomogeneous Korteweg de‐Vries Burger (i ‐KdVB) equations containing unique form of non‐uniform linear self‐consistent gradient‐driven sinks. A numerical illustrative scheme is procedurally constructed to examine the canonical fluctuations. It is seen that the eigenspectrum coevolves as electrostatic rarefactive damped oscillatory shock‐like structures and self‐gravitational compressive damped oscillatory shock‐like patterns . The irregular damping nature is attributable to the i ‐KdVB sinks. The aperiodicity in the hybrid rapid small downstream wavetrains is speculated to be deep‐rooted in the quasi‐linear gravito‐electrostatic interplay. The phase‐evolutionary dynamics grow as atypical non‐chaotic fixed‐point attractors . We, finally, indicate tentative astronomical applications relevant in large‐scale cosmic structure formation aboard facts and faults. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Hamiltonian structure of the Maxwell-Vlasov equations

Morrison [25] has observed that the Maxwell-Vlasov and Poisson-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. We derive another Poisson structure for these equations by using general methods of symplectic geometry. The main ingredients in our construction are the symplectic structure on the co-adjoint orbits for the group of canonical transformations, and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory. Our Poisson bracket satisfies the Jacobi identity, whereas Morrison's does not [37]. Our construction also shows where canonical variables can be found and can be applied to the Yang-Mills-Vlasov equations and to electromagnetic fluid dynamics.     

Symmetries and conservation laws for generalized hamiltonian systems

A class of dynamical systems which locally correspond to a general first-order system of Euler-Lagrange equations is studied on a contact manifold. These systems, called self-adjoint, can be regarded as generalizations of (time-dependent) Hamiltonian systems. It is shown that each one-parameter family of symmetries of the underlying contact form defines a parameter-dependent constant of the motion and vice versa. Next, an extension of the classical concept of canonical transformations is introduced. One-parameter families of canonical transformations are studied and shown to be generated as solutions of a self-adjoint system. Some of the results are illustrated on the Emden equation.     

Universal canonical entropy for gravitating systems

The thermodynamics of general relativistic systems with boundary, obeying a Hamiltonian constraint in the bulk, is determined solely by the boundary quantum dynamics, and hence by the area spectrum. Assuming, for large area of the boundary, (a) an area spectrum as determined by non-perturbative canonical quantum general relativity (NCQGR), (b) an energy spectrum that bears a power law relation to the area spectrum, (c) an area law for the leading order microcanonical entropy, leading thermal fluctuation corrections to the canonical entropy are shown to be logarithmic in area with a universal coefficient. Since the microcanonical entropy also has universal logarithmic corrections to the area law (from quantum space-time fluctuations, as found earlier) the canonical entropy then has a universal form including logarithmic corrections to the area law. This form is shown to be independent of the index appearing in assumption (b). The index, however, is crucial in ascertaining the domain of validity of our approach based on thermal equilibrium.     

Hamiltonian Structure for Dispersive and Dissipative Dynamical Systems

We develop a Hamiltonian theory of a time dispersive and dissipative inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The proposed Hamiltonian couples the given system to auxiliary fields, in the universal form of a so-called canonical heat bath. After integrating out the heat bath the original dissipative evolution is exactly reproduced. Furthermore, we show that the dynamics associated to a minimal Hamiltonian are essentially unique, up to a natural class of isomorphisms. Using this formalism, we obtain closed form expressions for the energy density, energy flux, momentum density, and stress tensor involving the auxiliary fields, from which we derive an approximate, “Brillouin-type,” formula for the time averaged energy density and stress tensor associated to an almost mono-chromatic wave.     

Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron

Mixed-mode dynamics is a complex type of dynamical behavior that has been observed both numerically and experimentally in numerous prototypical systems in the natural sciences. The compartmental Wilson-Callaway model for the dopaminergic neuron is an example of a system that exhibits a wide variety of mixed-mode patterns upon variation of a control parameter. One characteristic feature of this system is the presence of multiple time scales. In this article, we study the Wilson-Callaway model from a geometric point of view. We show that the observed mixed-mode dynamics is caused by a slowly varying canard structure. By appropriately transforming the model equations, we reduce them to an underlying three-dimensional canonical form that can be analyzed via a slight adaptation of the approach developed by M. Krupa, N. Popovic, and N. Kopell (unpublished).     

A Geometric Interpretation of the Second-Order Structure Function Arising in Turbulence

We primarily deal with homogeneous isotropic turbulence and use a closure model for the von Kármán-Howarth equation to study several geometric properties of turbulent fluid dynamics. We focus our attention on the application of Riemannian geometry methods in turbulence. Some advantage of this approach consists in exploring the specific form of a closure model for the von Kármán-Howarth equation that enables to equip a model manifold (a cylindrical domain in the correlation space) by a family of inner metrics (length scales of turbulent motion) which depends on time. We show that for large Reynolds numbers (in the limit of large Reynolds numbers) the radius of this manifold can be evaluated in terms of the second-order structure function and the correlation distance. This model manifold presents a shrinking cylindrical domain as time evolves. This result is derived by using a selfsimilar solution of the closure model for the von Kármán-Howarth equation under consideration. We demonstrate that in the new variables the selfsimilar solution obtained coincides with the element of Beltrami surface (or pseudo-sphere): a canonical surface of the constant sectional curvature equals − 1.      

Entropic Dynamics on Gibbs Statistical Manifolds

Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Here, we develop the entropic dynamics of a system, the state of which is described by a probability distribution. Thus, the dynamics unfolds on a statistical manifold that is automatically endowed by a metric structure provided by information geometry. The curvature of the manifold has a significant influence. We focus our dynamics on the statistical manifold of Gibbs distributions (also known as canonical distributions or the exponential family). The model includes an “entropic” notion of time that is tailored to the system under study; the system is its own clock. As one might expect that entropic time is intrinsically directional; there is a natural arrow of time that is led by entropic considerations. As illustrative examples, we discuss dynamics on a space of Gaussians and the discrete three-state system.     

Canonical methods for Hamiltonian systems: numerical experiments

A Hamiltonian system possesses dynamics (e.g. preservation of volume in phase space and symplectic structure) that call for special numerical integrators, namely canonical methods. Recent research on this aspect have shown that canonical numerical integrators may be needed for Hamiltonian systems. In this paper, we focus on numerical experiments that compare canonical and non-canonical numerical integrators. Test problems are taken from different areas in physical sciences. These experiments help to buttress the claims that canonical numerical integrators give results that mimic the qualitative behavior of the original system and that canonical numerical integrators are suitable for long time integrations. Our experiments indicate that higher-order canonical methods allow for larger timestep than lower-order canonical methods.     

An essential ambiguity of quantum theory

It is shown that there is a substantial family of Lagrangians for one and the same ground problem of Newtonian mechanics where linear momentum is conserved. These alternative Lagrangians involve an arbitrary function of conserved momentum, which reduces to a one-parameter family when it is required that the Galilean group be canonically represented. Then the canonical momentum becomes parameter-dependent: and this leads quantally to ambiguous commutation rules and Heisenberg uncertainty principles—for example the coordinate of one particle may be complementary not to its own mechanical momentum (as in standard treatments) but to that of a different particle. One possible way of resolving the ambiguity is by reformulating many-particle dynamics as higher-order one-particle (HOOP) dynamics by decoupling the standard lower-order equations of motion. This also admits relativistic generalization in which the Poincaré group is canonically represented.     

Retainability of canonical distributions for a Brownian particle controlled by a time-dependent harmonic potential

The retainability of canonical distributions for a Brownian particle controlled by a time-dependent harmonic potential is investigated in the overdamped and underdamped situations, respectively. Because of different time scales, the overdamped and underdamped Langevin equations (as well as the corresponding Fokker-Planck equations) lead to distinctive restrictions on protocols maintaining canonical distributions. Two special cases are analyzed in details: First, a Brownian particle is controlled by a time-dependent harmonic potential and embedded in medium with constant temperature; Second, a Brownian particle is controlled by a timedependent harmonic potential and embedded in a medium whose temperature is tuned together with the potential stiffness to keep a constant effective temperature of the Brownian particle. We find that the canonical distributions are usually retainable for both the overdamped and underdamped situations in the former case. However, the canonical distributions are retainable merely for the overdamped situation in the latter case. We also investigate general time-dependent potentials beyond the harmonic form and find that the retainability of canonical distributions depends sensitively on the specific form of potentials.