A Schwinger variational method for the Bloch equation

A Schwinger variational principle has been derived for use in quantum, manybody systems at finite temperatures. The variational principle is a stationary expression for the density matrix which may be iterated to improve an approximate density matrix. It also can be used to find stationary expressions for observables. If an approximate, parametrized density matrix is used, the parameters are varied to find the regions where the variational principle is stationary. The variational density matrix obtained with the optimal parameters can be regarded as optimal for that observable. The method has been applied to two model problems, a particle in a box and two hard spheres at finite temperatures. The advantages and shortcomings of the method are discussed.     

Large Deviations of Lattice Hamiltonian Dynamics Coupled to Stochastic Thermostats

We discuss the Donsker-Varadhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the Donsker-Varadhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary states as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling.     

Fermat principle for a nonstationary medium

One possible formulation of a variational principle of the Fermat type for systems with time-dependent parameters is suggested. In a stationary case, it reduces to the Mopertui-Lagrange least-action principle. A class of Hamiltonians (dispersion relations) is indicated, for which the variational principle reduces to the Fermat principle in a general nonstationary case. Hamiltonians that are homogeneous functions of momenta are in this category. For the important case of nondispersive waves (corresponding to Hamiltonians being homogeneous function of momenta order 1) the Fermat principle fully determines the geometry of the rays. Equations relating the variation of signal frequency with the rate of change of propagation time are established.     

A Technical Critique of Some Parts of the Free Energy Principle

We summarize the original formulation of the free energy principle and highlight some technical issues. We discuss how these issues affect related results involving generalised coordinates and, where appropriate, mention consequences for and reveal, up to now unacknowledged, differences from newer formulations of the free energy principle. In particular, we reveal that various definitions of the “Markov blanket” proposed in different works are not equivalent. We show that crucial steps in the free energy argument, which involve rewriting the equations of motion of systems with Markov blankets, are not generally correct without additional (previously unstated) assumptions. We prove by counterexamples that the original free energy lemma, when taken at face value, is wrong. We show further that this free energy lemma, when it does hold, implies the equality of variational density and ergodic conditional density. The interpretation in terms of Bayesian inference hinges on this point, and we hence conclude that it is not sufficiently justified. Additionally, we highlight that the variational densities presented in newer formulations of the free energy principle and lemma are parametrised by different variables than in older works, leading to a substantially different interpretation of the theory. Note that we only highlight some specific problems in the discussed publications. These problems do not rule out conclusively that the general ideas behind the free energy principle are worth pursuing.     

Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process

The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in nonequilibrium, namely for nonreversible systems. In this paper we consider a simple example of a nonequilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so-called quasi potential, which measures the cost of a fluctuation from the stationary state, is then defined by a variational problem for the dynamical large deviation rate function. By characterizing the optimal path, we prove that the quasi potential can also be obtained from a static variational problem introduced by Derrida, Lebowitz, and Speer.     

Relativistic non-Hamiltonian mechanics

Relativistic particle subjected to a general four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_μu^μ + c² = 0, where c is the speed of light in vacuum. In the general case, four-forces are non-potential, and the relativistic particle is a non-Hamiltonian system in four-dimensional pseudo-Euclidean space-time. We consider non-Hamiltonian and dissipative systems in relativistic mechanics. Covariant forms of the principle of stationary action and the Hamilton’s principle for relativistic mechanics of non-Hamiltonian systems are discussed. The equivalence of these principles is considered for relativistic particles subjected to potential and non-potential forces. We note that the equations of motion which follow from the Hamilton’s principle are not equivalent to the equations which follow from the variational principle of stationary action. The Hamilton’s principle and the principle of stationary action are not compatible in the case of systems with nonholonomic constraint and the potential forces. The principle of stationary action for relativistic particle subjected to non-potential forces can be used if the Helmholtz conditions are satisfied. The Hamilton’s principle and the principle of stationary action are equivalent only for a special class of relativistic non-Hamiltonian systems.     

Zur Begründung eines Variationsprinzipes für zerfallende Systeme

Taking into account the circumstance that the decay of an unstable microscopic system into two fragments is established by the counting of one of the decay products in a detector, the observed exponential decay law then asserts only knowledge of the spatiotemporal behaviour of the probability density (and therewith knowledge of the decaying state) at a large finite distance from the site of decay. We therefore formulate a variational principle, of which stationary functions show this decay behaviour. In addition to the resonant wave functions there are also solutions of the variational principle, which decrease exponentially with increasing distance, i.e., functions which could be used to describe the bound states. As the time-dependent treatment shows, the decaying states cannot occur in isolation in a scattering process. The mathematical characterisation of the decaying states via a variational principle is incorporated in a theory of open physical systems. In contradiction to the variational principle of Schrödinger our principle does not provide complete knowledge of the quantum states, but this is not needed in order to describe the decay.     

A variational approach to stochastic nonlinear problems

A variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities. Approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a Bernoulli oscillator driven by white noise. Comparison with exact data available for this system shows that the variational approach to such problems can be quite effective.     

Suspension Flows Over Countable Markov Shifts

We introduce the notion of topological pressure for suspension flows over countable Markov shifts, and we develop the associated thermodynamic formalism. In particular, we establish a variational principle for the topological pressure, and an approximation property in terms of the pressure on compact invariant sets. As an application we present a multifractal analysis for the entropy spectrum of Birkhoff averages for suspension flows over countable Markov shifts. The domain of the spectrum may be unbounded and the spectrum may not be analytic. We provide explicit examples where this happens. We also discuss the existence of full measures on the level sets of the multifractal decomposition.     

Occam’s Razor as a Formal Basis for a Physical Theory

We introduce the principle of Occam's Razor in a form that can be used as a basis for economical formulations of physics. This allows us to explain the general structure of the Lagrangian for a composite physical system, as well as some other artificial postulates behind the variational formulations of physical laws. As an example, we derive Hamilton's principle of stationary action together with the Lagrangians for the cases of Newtonian mechanics, relativistic mechanics and a relativistic particle in an external gravitational field.     

Variational principle for flow of non-Newtonian fluids

A variational principle is formulated for finding stationary solutions of the equations of motion for an incompressible non-Newtonian fluid with constitutive equation of the Reiner-Rivlin type. The basic functional is related, but not identical to the rate of energy dissipation. It can be used for analyzing the stability of the stationary state against small perturbations.     

Approximation methods for stationary solutions of discrete master equations

Two methods for finding stationary solutions of discrete master equation in general cases without detailed balance are set up and compared for the case of the Laser equation. The first method starts from a reduced master equation obtained for appropriate forms of the Liouvillean, the second from a variational principle for the stationary solution of the full master equation. In the Laser case, both methods lead to forms of the stationary solution essentially equivalent to the result of Dohm.     

A new type of adiabatic invariants for disturbed Birkhoffian system of Herglotz type

The generalized variational principle of Herglotz type provides an effective way to study the problems of conservative and non-conservative systems in a unified way. According to the differential variational principle of Herglotz type, we study the adiabatic invariants for a disturbed Birkhoffian system in this paper. Firstly, the differential equations of motion of the Birkhoffian system based upon this variational principle are given, and the exact invariant of Herglotz type of the system is introduced. Secondly, a new type of adiabatic invariants for the system under the action of small perturbation is obtained. Thirdly, the inverse theorem of adiabatic invariant for the disturbed Birkhoffian system of Herglotz type is obtained. Finally, an example is given.     

NOETHER'S THEOREM OF NONHOLONOMIC SYSTEMS OF NON-CHETAEV'S TYPE WITH UNILATERAL CONSTRAINTS

In this paper, we present Noether's theorem and its inverse theorem for nonholonomic systems of non-Chetaev's type with unilateral constraints. We present first the principle of Jourdain for the system and, on the basis of the invariance of the differential variational principle under the infinitesimal transformations of groups, we have established Noether's theory for the above systems. An example is given to illustrate the application of the result.     

Dirac structures in Lagrangian mechanics Part II: Variational structures

Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler–Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton–Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange–d’Alembert principle called the Lagrange–d’Alembert–Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an $L$–$C$ circuit are given to illustrate the results.     

Birkhoff系统的时间积分定理

研究Birkhoff系统的时间积分定理.建立Birkhoff系统的时间积分等式，由此等式导出系统的类功率方程，Pfaff-Birkhoff积分变分原理以及Pfaff-Birkhoff-d′Alembert微分变分原理. 关键词： Birkhoff系统 时间积分等式 类功率方程 变分原理     

A Fractional Schrödinger Equation and Its Solution

This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.     

Stochastic variational derivations of the Navier-Stokes equation

Stochastic calculus of variations in terms of sample paths is shown to be equivalent to ordinary calculus of variations in terms of local drift fields when applied to Markov processes. This equivalence enables us to derive the Navier-Stokes equation directly through a variational principle. Partially supported by the Swiss National Science Foundation.