Table of contents

Recently the author used an information theoretical formulation of non-equilibrium statistical mechanics (MaxEnt) to derive the fluctuation theorem (FT) concerning the probability of second law violating phase-space paths. A less rigorous argument leading to the variational principle of maximum entropy production (MEP) was also given. Here a more rigorous and general mathematical derivation of MEP from MaxEnt is presented, and the relationship between MEP and the FT is thereby clarified. Specifically, it is shown that the FT allows a general orthogonality property of maximum information entropy to be extended to entropy production itself, from which MEP then follows. The new derivation highlights MEP and the FT as generic properties of MaxEnt probability distributions involving anti-symmetric constraints, independently of any physical interpretation. Physically, MEP applies to the entropy production of those macroscopic fluxes that are free to vary under the imposed constraints, and corresponds to selection of the most probable macroscopic flux configuration. In special cases MaxEnt also leads to various upper bound transport principles. The relationship between MaxEnt and previous theories of irreversible processes due to Onsager, Prigogine and Ziegler is also clarified in the light of these results.

The time evolution of a coagulating mixture in which binary coagulation governs the temporal changes to the particle mass and composition spectra is studied under the assumption that the coagulation kernel is proportional to the product of masses of the first and the second components in the coalescing particles. This model is shown to reveal the sol–gel transition, i.e., the formation of a cluster with mass comparable to the total mass of the whole system. This letter reports on the exact solution of this model. The evolution equation for the generating functional defining all properties of coagulating systems is solved exactly for this particular kernel. The single-particle mass spectrum is analysed in the thermodynamic limit and it is demonstrated how the gel appears in this mixed system.

We construct two pairs of quasicoordinates and quasimomenta in a finite phase plane, which form sets of conjugate variables. In such a plane the coordinate x is quantized with a step c, and the momentum p with a step , where Mc is the size of the phase plane in the x-direction. The construction depends crucially on the possibility of writing M = M₁M₂ with M₁ and M₂ relatively prime. The conjugate variables are applied to Harper-like Hamiltonians. It is shown how to design physical systems with energy spectra containing any desired number of discrete energy levels, say M₁, each of them having a prescribed degeneracy M₂.

Recently (Samsonov B F and Roy P 2005 J. Phys. A: Math. Gen.38 L249) an explicit example of a -symmetric non-diagonalizable Hamiltonian has been given. In this letter we show that such Hamiltonians appear as supersymmetric (SUSY) partners of Hermitian (hence diagonalizable) Hamiltonians and can be transformed back to diagonalizable forms by appropriate SUSY transformations.

We study the generating function of the spin–spin correlation functions in the ground state of the anti-ferromagnetic spin-1/2 Heisenberg chain without magnetic field. We have found its fundamental functional relations from those for general correlation functions, which originate in the quantum Knizhnik–Zamolodchikov equation. Using these relations, we have calculated the explicit form of the generating functions up to n = 6. Accordingly we can obtain the spin–spin correlator ⟨S^z_jS^z_j+k⟩ up to k = 5.

Using Monte Carlo simulations on different system sizes we determine with high precision the critical thresholds of two families of directed percolation models on a square lattice. The thresholds decrease exponentially with the degree of connectivity. We conjecture that p_c decays exactly as the inverse of the coodination number.

We revisit the Jordan–Wigner transformation, showing that—rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators—it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan–Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies.

We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by the use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting of the maps. Although these trees have no simple local characterization, we prove that their enumeration may be performed upon introducing a larger class of 'admissible' trees with possibly doubly occupied edges and summing them with appropriate signed weights. The proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non-sectile edges. The admissible trees are characterized by simple local rules, allowing eventually for an exact enumeration of planar bicubic maps with hard particles. We also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re-derive the enumeration of quartic maps with Ising spins in the present framework of admissible trees. We finally comment on a possible interpretation in terms of branching processes.

We propose an improved parametric form for the equation of state of three-dimensional O(N) spin systems. The proposed form is a series expansion with two sets of terms, which contribute (mainly) separately to the description of the high- and low-temperature regions of the phase diagram. Our goal is a better description of the low-temperature phase at zero magnetic field (i.e. the coexistence line), characterized by singularities induced by Goldstone modes. We test our proposed form by comparison with existing Monte Carlo data for the N = 4 case, which is of interest in studies of the QCD phase transition and for which the Goldstone-mode effects are quite pronounced. We find that the description of the numerical equation of state is indeed improved with respect to other fitting forms. In all cases considered we determine the coefficients nonperturbatively, from fits to the data. As a consequence, we are able to obtain a very precise characterization of the pseudo-critical line for the model.

We analytically study the one-dimensional asymmetric simple exclusion process with open boundaries under sublattice-parallel updating scheme. We investigate the stationary state properties of this model conditioned on finding a given particle number in the system. Recent numerical investigations have shown that the model possesses three different phases in this case. Using a matrix product method we calculate both the exact canonical partition function and also density profiles of the particles in each phase. Application of the Yang–Lee theory reveals that the model undergoes two second-order phase transitions at critical points. These results confirm the correctness of our previous numerical studies.

We study the distribution of cycles of length h in large networks (of size N ≫ 1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h_∗ ∼ N^α. Our results suggest that h_∗ and the exponent α might usefully characterize broad families of networks. In addition to an exact counting of cycles in hierarchical nets, we present a Monte Carlo sampling algorithm for approximately locating h_∗ and reliably determining α. Our empirical results indicate that for small random scale-free nets of degree exponent λ, α = 1/(λ − 1), and α grows as the nets become larger.

We provide all Lie point symmetries of a system of partial differential equations, of great interest to plasma physics; we deduce the corresponding invariant solutions and discuss their physical interpretation. We also consider some of its conditional and partial symmetries and obtain other interesting solutions. Finally, we point out that, although the system can be cast in divergence form and admits conserved currents, it does not admit potential symmetries.

In the model suggested by Smilansky (2004 Waves Random Media14 143–53) one studies an operator describing the interaction between a quantum graph and a system of K one-dimensional oscillators attached at several different points in the graph. The present paper is the first one in which the case K > 1 is investigated. For the sake of simplicity, we consider K = 2, but our argument is of a general character. In this first of two papers on the problem, we describe the absolutely continuous spectrum. Our approach is based upon scattering theory.

We present the role of invariants in obtaining exact solutions of differential equations. Firstly, conserved vectors of a partial differential equation (p.d.e.) allow us to obtain reduced forms of the p.d.e. for which some of the Lie point symmetries (in vector field form) are easily concluded and, therefore, provide a mechanism for further reduction. Secondly, invariants of reduced forms of a p.d.e. are obtainable from a variational principle even though the p.d.e. itself does not admit a Lagrangian. In this latter case, the reductions carry all the usual advantages regarding Noether symmetries and double reductions. The examples we consider are nonlinear evolution-type equations such as the Korteweg–deVries equation, but a detailed analysis is made on the Fisher equation (which describes reaction–diffusion waves in biology, inter alia). Other diffusion-type equations lend themselves well to the method we describe (e.g., the Fitzhugh Nagumo equation, which is briefly discussed). Some aspects of Painlevé properties are also suggested.

We show that a series of recurrent inequalities derived in N = 3 have the same formal expressions in any dimension N ⩾ 2. They are derived from the multipole sum rules, and provide us with upper bounds for the moments of the ground-state density depending only on the lowest multipole excitation energy. These bounds are transformed into approximate recurrent relations by means of an empirical correction factor. The 1/r potential and the harmonic oscillator play a key role in establishing this factor, which is exact for these two potentials by construction. For a large class of potentials, we show that this factor tends to 1 as N → ∞. In such cases, at the large-N limit, the lowest state for each multipole excitation exhausts the sum rule. It thus acquires the characteristics of the one-phonon excitation typical of the harmonic oscillator.

We consider a semiclassical approximation, first derived by Heller and coworkers, for the time evolution of an originally Gaussian wave packet in terms of complex trajectories. We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller's thawed Gaussian approximation. The other approximations are non-Gaussian and may involve several trajectories determined by mixed initial–final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive Gaussian potential and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-Gaussian approximations with real trajectories work well in some cases, whereas the thawed Gaussian works only in very simple situations.

We consider the self-adjoint extensions (SAE) of the symmetric supercharges and Hamiltonian for a model of SUSY quantum mechanics in with a singular superpotential. We show that only for two particular SAE, whose domains are scale invariant, the algebra of N = 2 SUSY is realized, one with manifest SUSY and the other with spontaneously broken SUSY. Otherwise, only the N = 1 SUSY algebra is obtained, with spontaneously broken SUSY and non-degenerate energy spectrum.

Two schemes of the quantization of dynamical variables of dissipative systems are studied with an example of a two-level atom interacting with phonons scattered from an environment. Transformations of relevant Fock-space operators or rigged-Fock-space operators are proposed in order to diagonalize their equations of motion. It is shown that the damping of dispersionless phonons leads to the formation of a quasi-stationary state of the system—a polaron, which is found in both approaches. A time dependence of the polaron energy and the spectral lineshape of the two-level subsystem are evaluated. A transformation diagonalizing the equations of motion of the relevant classical problem (of a harmonic oscillator coupled to a field of damped waves) is found to be similar to the quantum case.

The Schrödinger equation in integral form is applied to the one-dimensional scattering problem in the case of a narrow potential barrier. Since the kernel can be considered, in a first approximation, separable, an explicit expression for the propagator is found by means of the complementary error function. The problem of a particle confined in a half-space and interacting with a narrow potential barrier is also considered and solved in an approximate way.

Some general properties of a bifurcated oscillator potential in one dimension are analysed. Appropriate wavefunctions in different regions with continuity conditions lead to a simple relation for the energies. The properties of these energies for small and large values of separation are used to develop simple and accurate expressions for the energies of all the states for all values of separation.

Using the intertwining relation we construct a pseudosuperpartner for a (non-Hermitian) Dirac-like Hamiltonian describing a two-level system interacting in the rotating wave approximation with the electric component of an electromagnetic field. The two pseudosuperpartners and pseudosupersymmetry generators close a quadratic pseudosuperalgebra. A class of time-dependent electric fields for which the equation of motion for a two-level system placed in this field can be solved exactly is obtained. A new interesting phenomenon is observed. There exists a time-dependent detuning of the field frequency from the resonance value such that the probability to populate the excited level ceases to oscillate and becomes a monotonically growing function of time tending to 3/4. It is shown that near this fixed excitation regime the probability exhibits two kinds of oscillations. The oscillations with small amplitude and frequency close to the Rabi frequency (fast oscillations) take place at the background of those with big amplitude and small frequency (slow oscillations). During the period of slow oscillations, the minimal value of the probability to populate the excited level may exceed 1/2, suggesting for an ensemble of such two-level atoms the possibility of acquiring an inverse population and exhibit lasing properties.

An exact solution of the Dirac equation for a particle whose potential energy and mass are inversely proportional to the distance from the force centre has been found. The bound states exist provided the length scale a which appears in the expression for the mass is smaller than the classical electron radius e²/mc². Furthermore, bound states also exist for negative values of a even in the absence of the Coulomb interaction. In first order perturbation theory the corrections to the energy caused by the terms of dipole nature in the expression for the mass are calculated. Quasirelativistic expansion of the energy has been carried out, and a modified expression for the fine structure of the energy levels has been obtained. The problem of kinetic energy operator in the Schrödinger equation is discussed for the case of position-dependent mass. In particular, we have found that for highly excited states the mutual ordering of the inverse mass and momentum operator in the non-relativistic theory is not important.

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C₁ coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A separable second-order hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat-kernel expansion which are evaluated explicitly.