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Journal of Physics A: Mathematical and Theoretical - latest papers

Latest articles for Journal of Physics A: Mathematical and Theoretical

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  • Nonuniform Bose–Einstein condensate: I. An improvement of the Gross–Pitaevskii method
    A nonuniform condensate is usually described by the Gross–Pitaevskii (GP) equation, which is derived with the help of the c-number ansatz . Proceeding from a more accurate operator ansatz , where N is the number of Bose particles, we find the equation , which we call the GPN equation. It differs from the GP equation by the factor . We compare the accuracy of the GP and GPN equations by analysing the ground state of a one-dimensional system of point bosons with repulsive interaction (c > 0) and zero boundary conditions. Both equations are solved numerically, and the system energy E and the particle density profile are determined for the mean particle density , different values of N and of the coupling constant . The solutions are compared with the exact ones obtained by the Bethe ansatz. The results show that the GP and GPN equations equally well describe the many-particle system ( ) being in the weak coupling regime ( ). But for the few-boson system ( ) with the solutions of the GPN equation are in much better agreement with the exact ones. Thus, the multiplier allows one to describe few-boson systems with high accuracy. This means that it is reasonable to extend the notion of Bose–Einstein condensation to few-particle systems.

  • Minimum time connection between non-equilibrium steady states: the Brownian gyrator
    We study the problem of minimising the connection time between non-equilibrium steady states of the Brownian gyrator. This is a paradigmatic model in non-equilibrium statistical mechanics, an overdamped Brownian particle trapped in a two-dimensional elliptical potential, with the two degrees of freedom (x, y) coupled to two, in principle different, thermal baths with temperatures Tx and Ty, respectively. Application of Pontryagin’s Maximum Principle reveals that shortest protocols belong to the boundaries of the control set defined by the limiting values of the parameters (k, u) characterising the elliptical potential. We identify two classes of optimal minimum time protocols, i.e. brachistochrones: (i) regular bang–bang protocols, for which (k, u) alternatively take their minimum and maximum values allowed, and (ii) infinitely degenerate singular protocols. We thoroughly investigate the minimum connection time over the brachistochrones in the limit of having infinite capacity for compression. A plethora of striking phenomena emerge: sets of states attained at null connection times, discontinuities in the connection time along adjacent target states, and the fact that, starting from a state in which the oscillators are coupled, uncoupled states are impossible to reach in a finite time.

  • Quantizations of transposed Poisson algebras by Novikov deformations
    The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson algebras, and hence the Novikov deformation is defined to be the quantization of the corresponding transposed Poisson algebra. As a direct consequence, we revisit the relationship between transposed Poisson algebras and Novikov–Poisson algebras due to the fact that there is a natural Novikov deformation of the commutative associative algebra in a Novikov–Poisson algebra. Hence all transposed Poisson algebras of Novikov–Poisson type, including unital transposed Poisson algebras, can be quantized. Finally, we classify the quantizations of 2-dimensional complex transposed Poisson algebras in which the Lie brackets are non-abelian up to equivalence.

  • Nonuniform Bose–Einstein condensate: II. Doubly coherent states
    We find stationary excited states of a one-dimensional system of N spinless point bosons with repulsive interaction and zero boundary conditions by numerically solving the time-independent Gross–Pitaevskii equation. The solutions are compared with the exact ones found in the Bethe-ansatz approach. We show that the jth stationary excited state of a nonuniform condensate of atoms corresponds to a Bethe-ansatz solution with the quantum numbers . On the other hand, such values of correspond to a condensate of N elementary excitations (the Bogoliubov quasiparticles) with the quasimomentum , where L is the system size. Thus, each stationary excited state of the condensate is ‘doubly coherent’, since it corresponds simultaneously to a condensate of N atoms and a condensate of N elementary excitations. We find the energy E and the particle density profile for such states. The possibility of experimental production of these states is also discussed.

  • Non-existence of causal, standard classical electrodynamics with point charged particle
    The existence of consistent, standard and causal theory of point charged particle (for example electron) interacting with electromagnetic field was the subject of many investigations. This problem is often stated as the description of the system of electromagnetic pulse of radiation interacting with single point charged particle (for example electron). The correct theory should give the causal particle trajectory for any electromagnetic pulse of radiation. Up to now no such theory was formulated. We show that for certain electromagnetic pulse of radiation and point particle being initially at rest, there does not exist a causal and physical particle trajectory that satisfies energy and momentum conservation. This shows that the causal, standard electrodynamics of scalar point particles, valid for all possible external pulses, does not exist.