A local approach to dimensional reduction: I. General formalism

We present a formalism for dimensional reduction based on the local properties of invariant cross-sections (“fields”) and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact.

In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e. to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a “non-canonical” reduction.

In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space.     

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system

The Darboux–Egoroff system of PDEs with any number $n\ge 3$ of independent variables plays an essential role in the problems of describing $n$-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux–Egoroff system and describe some symmetries generated by these operators.The constructed recursion operators are not pseudodifferential, but are Bäcklund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux–Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Pobořil, and Sergyeyev.In the structure of the obtained third and fifth order symmetries of the Darboux–Egoroff system, one finds the third and fifth order flows of an $(n-1)$-component vector modified KdV hierarchy.The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental–van de Leur twisted loop group action on the space of semisimple Frobenius manifolds.We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.     

A Geometrical Approach to Hojman Theorem of a Rotational Relativistic Birkhoffian System

A geometrical approach to the Hojman theorem of a rotational relativistic Birkhoffian system is presented.The differential equations of motion of the system are established. According to the invariance of differential equations under infinitesimal transformation, the determining equations of Lie symmetry are constructed. A new conservation law of the system, called Hojman theorem, is obtained, which is the generalization of previous results given sequentially by Hojman, Zhang, and Luo et al. In terms of the theory of modern differential geometry a proof of the theorem is given.     

A Geometrical Approach to Hojman Theorem of a Rotational Relativistic Birkhoffian System

A geometrical approach to the Hojman theorem of a rotational relativistic Birkhoffian system is presented.The differential equations of motion of the system are established. According to the invariance of differential equations under infinitesimal transformation, the determining equations of Lie symmetry are constructed. A new conservation law of the system, called Hojman theorem, is obtained, which is the generalization of previous results given sequentially by Hojman, Zhang, and Luo et al. In terms of the theory of modern differential geometry a proof of the theorem is given.     

Relaxation of Collisional Magnetically Confined Plasmas to Mechanical Equilibria

The relaxation of magnetically confined plasmas in a toroidal geometry is analyzed. From the equations for the Hermitian moments, we show how the system relaxes towards the mechanical equilibrium. In the space of the parallel generalized frictions, after fast transients, the evolution of collisional magnetically confined plasmas is such that the projections of the evolution equations for the parallel generalized frictions and the shortest path on the Hermitian moments coincide. For spatially‐extended systems, a similar result is valid for the evolution of the thermodynamic mode (i.e., the mode with wave‐number k = 0 ). The expression for the affine connection of the space covered by the generalized frictions, close to mechanical equilibria, is also obtained. The knowledge of the components of the affine connection is a fundamental prerequisite for the construction of the (nonlinear) closure theory on transport processes (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantum statistical monte carlo methods and applications to spin systems

A short review is given concerning the quantum statistical Monte Carlo method based on the equivalence theorem⁽¹⁾ thatd-dimensional quantum systems are mapped onto (d+1)-dimensional classical systems. The convergence property of this approximate tansformation is discussed in detail. Some applications of this geneal appoach to quantum spin systems are reviewed. A new Monte Carlo method, “thermo field Monte Carlo method,” is presented, which is an extension of the projection Monte Carlo method at zero temperature to that at finite temperatures. Invited talk presented at “Frontiers of Quantum Monte Carlo,” Los Alamos National Laboratory, September 3–6, 1985.     

Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation

We geometrically study the Legendre duality relation that plays an important role in statistical physics with the standard or generalized entropies. For this purpose, we introduce dualistic structure defined by information geometry, and discuss concepts arising in generalized thermostatistics, such as relative entropies, escort distributions and modified expectations. Further, a possible generalization of these concepts in a certain direction is also considered. Finally, as an application of such a geometric viewpoint, we briefly demonstrate several new results on the behavior of the solution to a nonlinear diffusion equation called the porous medium equation.