Narrow Results

In this note, we consider the mathematical aspects of the functional mechanics proposed and developed by I. V. Volovich. In particular, we briefly review the Liouville equation used to describe the dynamics of wave packets, study the characteristics of wave packets, such as the dynamics of their averages and dispersions, the features of dynamics in the case of Hamiltonian and Lagrangian theory, and give a number of simple examples illustrating general constructions.

We consider dynamical systems that are described by fractional power of coordinates and momenta. The fractional powers can be considered as a convenient way to describe systems in the fractional dimension space. For the usual space the fractional systems are non-Hamiltonian. Generalized transport equation is derived from Liouville and Bogoliubov equations for fractional systems. Fractional generalization of average values and reduced distribution functions are defined. Gasdynamic equations for fractional systems are derived from the generalized transport equation.

A Lie algebra which consists of linear combinations of one basis of the Lie algebra A₁ is presented for which an isospectral Lax pair is exhibited. By using the zero curvature equation, the generalized mKdV equation, Liouville equation and sine-Gordon equation, sinh-Gordon equation are generated via polynomial expansions. Finally, we investigate a kind of formal Bäcklund transformation for the generalized sine-Gordon equation. The explicit Bäcklund transformation of the standard sine-Gordon equation is presented. The other equations given in the paper are obtained similarly.

We derive the canonical distribution as a stationary solution of the Liouville equation for the classical dissipative system. Dissipative classical systems can have stationary states that look like canonical Gibbs distributions. The condition for non-potential forces which leads to this stationary solution is very simple: the power of the non-potential forces must be directly proportional to the velocity of the Gibbs phase (phase entropy density) change. The example of the canonical distribution for a linear oscillator with friction is considered.

The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Fractional kinetic equation for the system of charged particles are considered.

We consider a sequence of blowup solutions of a two-dimensional, second-order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci–Chen–Lin–Tarantello, it is proved that the profile of the solutions differs from global solutions of a Liouville-type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.

We discuss compactness, blow-up and quantization phenomena for the prescribed Q-curvature equation (-Δ)^m u_k = V_ke^2mu_k on open domains of ℝ^2m. Under natural integral assumptions we show that when blow-up occurs, up to a subsequence

Let $m\ge 2$ be an integer. For any open domain $\mathrm{\Omega }\subset {ℝ}^{2m}$, non-positive function $\phi \in C^{\infty }(\mathrm{\Omega })$ such that ${\mathrm{\Delta }}^m\phi \equiv 0$, and bounded sequence $(V_k)\subset L^{\infty }(\mathrm{\Omega })$ we prove the existence of a sequence of functions $(u_k)\subset C^{2m-1}(\mathrm{\Omega })$ solving the Liouville equation of order $2m$

Assume $\mathrm{\Omega }\subset {ℝ}^2$ is a planar domain, and $u$ is a locally bounded distributional solution to the elliptic equation

Let $\mathrm{\Omega }\subset {ℝ}^3$ be an open connected domain and $\mathrm{\Delta }$ stand for the Laplacian in ${ℝ}^3$. For a uniformly bounded sequence $(Q_k)\subset L^{\infty }(\mathrm{\Omega })$ having a fixed sign and non-positive bi-harmonic functions $\mathrm{\Phi }(x,t)\in C^{\infty }({ℝ}_{+}^4\cup \mathrm{\Omega })$, we prove that there exists a sequence of solutions $(u_k)$ to

The Liouville equation differs from the von Neumann equation "only" by a characteristic superoperator. We demonstrate this for Hamiltonian dynamics, in general, and for the Jaynes-Cummings model, in particular. Employing superspace (instead of Hilbert space), we describe time evolution of density matrices in terms of path integrals, which are formally identical for quantum and classical mechanics. They only differ by the interaction contributing to the action. This allows us to import tools developed for Feynman path integrals, in order to deal with superoperators instead of quantum mechanical commutators in real time evolution. Perturbation theory is derived. Besides applications in classical statistical physics, the "classical path integral" and the parallel study of classical and quantum evolution indicate new aspects of (dynamically assisted) entanglement (generation). Our findings suggest to distinguish intra- from inter-space entanglement.

The paper aims at clarifying the essential relationship between traditional probability density evolution equations and the generalized probability density evolution equation which is developed by the authors in recent years. Using the principle of preservation of probability as a uniform fundamental, the probability density evolution equations, including the Liouville equation, Fokker–Planck equation and the Dostupov–Pugachev equation, are derived from the physical point of view. It is pointed out that combining with Eulerian or Lagrangian description of the associated dynamical system will lead to different probability density evolution equations. Particularly, when both the principle and dynamical systems are viewed from Lagrangian description, we are led to the generalized probability density evolution equation.

Some elements of classical mechanics and classical statistical mechanics are formulated in terms of fibre bundles. In the bundle approach the dynamical and distribution functions are replaced by liftings of paths in a suitably chosen bundle. Their time evolution is described by appropriate linear transports along paths in it or, equivalently, by corresponding invariant bundle equations of motion. In particular, the bundle version of the Liouville equation is derived.

The Boltzmann–Enskog kinetic equation is known to describe the irreversible dynamics of a dilute hard sphere gas. From the other side, the Boltzmann–Enskog equation is known to have reversible microscopic solutions. We show that the reversibility or irreversibility property of the Boltzmann–Enskog equation depends on the considered class of solutions. If the considered class solutions is sums of delta-functions, then the equation is reversible. If the considered class of solutions is continuous functions, then the equation is irreversible. We suggest a new way of the reconciliation of the irreversible kinetic equation with the reversible particle dynamics. We postulate the Boltzmann–Enskog equation and regard its microscopic solutions as a kind of derivation of the microscopic equations of motion from the Boltzmann–Enskog equation. From this point of view, the “fundamental” laws of motion are not the laws for individual particles (reductionism), but the laws for the single-particle density function, which describes the system as a whole (holism). Laws of motion for the individual particles are approximations. This work can be regarded as a development of recently proposed functional mechanics, which is aimed to make mechanics more appropriate for applications to complex and biological systems.