Francisco J. Sevilla

The BCS–Bose crossover theory

Abstract We contrast four distinct versions of the BCS–Bose statistical crossover theory according to the form assumed for the electron–number equation that accompanies the BCS gap equation. The four versions correspond to explicitly accounting for two-hole-(2h) as well as two-electron-(2e) Cooper pairs (CPs), or both in equal proportions, or only either kind. This follows from a recent generalization of the Bose–Einstein condensation (GBEC) statistical theory that includes not boson–boson interactions but rather 2e- and also (without loss of generality) 2h-CPs interacting with unpaired electrons and holes in a single-band model that is easily converted into a two-band model. The GBEC theory is essentially an extension of the Friedberg–Lee 1989 BEC theory of superconductors that excludes 2h-CPs. It can thus recover, when the numbers of 2h- and 2e-CPs in both BE-condensed and non-condensed states are separately equal, the BCS gap equation for all temperatures and couplings as well as the zero-temperature BCS (rigorous-upper-bound) condensation energy for all couplings. But ignoring either 2h- or 2e-CPs it can do neither. In particular, only half the BCS condensation energy is obtained in the two crossover versions ignoring either kind of CPs. We show how critical temperatures T c from the original BCS–Bose crossover theory in 2D require unphysically large couplings for the Cooper/BCS model interaction to differ significantly from the T c s of ordinary BCS theory (where the number equation is substituted by the assumption that the chemical potential equals the Fermi energy).

A generalized master equation approach to modelling anomalous transport in animal movement

We present some models of random walks with internal degrees of freedom that have the potential to find application in the context of animal movement and stochastic search. The formalism we use is based on the generalized master equation which is particularly convenient here because of its inherent coarse-graining procedure whereby a random walker position is averaged over the internal degrees of freedom. We show some instances in which non-local jump probabilities emerge from the coupling of the motion to the internal degrees of freedom, and how the tuning of one parameter can give rise to sub-, super- and normal diffusion at long times. Remarks on the relation between the generalized master equation, continuous time random walks and fractional diffusion equations are also presented.

Phase transitions induced by complex nonlinear noise in a system of self-propelled agents

We propose a comprehensive dynamical model for cooperative motion of self-propelled particles, e.g., flocking, by combining well-known elements such as velocity-alignment interactions, spatial interactions, and angular noise into a unified Lagrangian treatment. Noise enters into our model in an especially realistic way: it incorporates correlations, is highly nonlinear, and it leads to a unique collective behavior. Our results show distinct stability regions and an apparent change in the nature of one class of noise-induced phase transitions, with respect to the mean velocity of the group, as the range of the velocity-alignment interaction increases. This phase-transition change comes accompanied with drastic modifications of the microscopic dynamics, from nonintermittent to intermittent. Our results facilitate the understanding of the origin of the phase transitions present in other treatments.

Low-dimensional BEC

The Bose-Einstein condensation (BEC) temperature Tc of Cooper pairs (CPs) created from a general interfermion interaction is determined for a linear, as well as the usually assumed quadratic, energy vs center-of-mass momentum dispersion relation. This explicit Tc is then compared with a widely applied implicit one of Wen & Kan (1988) in d=2+∈ dimensions, for small ∈, for a geometry of an infinite stack of parallel (e.g., copperoxygen) planes as in, say, a cuprate superconductor, and with a new result for linear-dispersion CPs. The implicit formula gives Tc values only slightly lower than those of the explicit formula for typical cuprate parameters.

Smoluchowski diffusion equation for active Brownian swimmers.

We study the free diffusion in two dimensions of active Brownian swimmers subject to passive fluctuations on the translational motion and to active fluctuations on the rotational one. The Smoluchowski equation is derived from a Langevin-like model of active swimmers and analytically solved in the long-time regime for arbitrary values of the Péclet number; this allows us to analyze the out-of-equilibrium evolution of the positions distribution of active particles at all time regimes. Explicit expressions for the mean-square displacement and for the kurtosis of the probability distribution function are presented and the effects of persistence discussed. We show through Brownian dynamics simulations that our prescription for the mean-square displacement gives the exact time dependence at all times. The departure of the probability distribution from a Gaussian, measured by the kurtosis, is also analyzed both analytically and computationally. We find that for the inverse of Péclet numbers ≲0.1, the distance from Gaussian increases as ∼t(-2) at short times, while it diminishes as ∼t(-1) in the asymptotic limit.

Cooper pairs as bosons

Although BCS pairs of fermions are known to obey neither Bose–Einstein (BE) commutation relations nor BE statistics, we show how Cooper pairs (CPs), whether the simple original ones or the CPs recently generalized in a many-body Bethe–Salpeter approach, being clearly distinct from BCS pairs at least obey BE statistics. Hence, contrary to widespread popular belief, CPs can undergo BE condensation to account for superconductivity if charged, as well as for neutral-atom fermion superfluidity where CPs, but uncharged, are also expected to form.

Dimensional crossover of a boson gas in multilayers

We obtain the thermodynamic properties for a noninteracting Bose gas constrained on multilayers modeled by a periodic Kronig-Penney delta potential in one direction and allowed to be free in the other two directions. We report Bose-Einstein condensation (BEC) critical temperatures, chemical potential, internal energy, specific heat, and entropy for different values of a dimensionless impenetrability P{>=}0 between layers. The BEC critical temperature T{sub c} coincides with the ideal gas BEC critical temperature T{sub 0} when P=0 and rapidly goes to zero as P increases to infinity for any finite interlayer separation. The specific heat C{sub V} as a function of absolute temperature T for finite P and plane separation a exhibits one minimum and one or two maxima in addition to the BEC, for temperatures larger than that of BEC T{sub c}. This highlights the effects due to particle confinement. We then discuss a distinctive dimensional crossover of the system through the specific heat behavior driven by the magnitude of P. For T T{sub c}, it is exhibited by a broad minimum in C{sub V}(T).

Bose-Einstein Condensation in Multilayers

The critical BEC temperature Tc of a non interacting boson gas in a layered structure like those of cuprate superconductors is shown to have a minimum Tc,m, at a characteristic separation between planes am. It is shown that for a<am, Tc increases monotonically back up to the ideal Bose gas T0 suggesting that a reduction in the separation between planes, as happens when one increases the pressure in a cuprate, leads to an increase in the critical temperature. For finite plane separation and penetrability the specific heat as a function of temperature shows two novel crests connected by a valley in addition to the well-known BEC peak at Tc associated with the 3D behavior of the gas. For completely impenetrable planes the model reduces to many disconnected infinite slabs for which just one hump survives becoming a peak only when the slab widths are infinite.

Harmonically trapped ideal quantum gases

Abstract:We solve the problem of a Bose or Fermi gas in d-dimensions trapped by δ ⩽ d mutually perpendicular harmonic oscillator potentials. From the grand potential we derive their thermodynamic functions (internal energy, specific heat, etc.) as well as a generalized density of states. The Bose gas exhibits Bose-Einstein condensation at a nonzero critical temperature Tc if and only if d + δ > 2, along with a jump in the specific heat at Tc if and only if d + δ > 4. Specific heats for both gas types precisely coincide as functions of temperature when d + δ = 2. The trapped system behaves like an ideal free quantum gas in d + δ dimensions. For δ = 0 we recover all known thermodynamic properties of ideal quantum gases in d dimensions, while in 3D for δ = 1, 2 and 3 one simulates behavior reminiscent of quantum wells, wires anddots, respectively. Good agreement is found between experimental critical temperatures for the trapped boson gases 3787Rb, 37Li, 3785Rb, 24He, 1941K and the known theoretical expression which is a special case for d = δ = 3, but only moderate agreement for 1127Na and 11H.

Emergence of Collective Motion in a Model of Interacting Brownian Particles.

By studying a system of Brownian particles that interact among themselves only through a local velocity-alignment force that does not affect their speed, we show that self-propulsion is not a necessary feature for the flocking transition to take place as long as underdamped particle dynamics can be guaranteed. Moreover, the system transits from stationary phases close to thermal equilibrium, with no net flux of particles, to far-from-equilibrium ones exhibiting collective motion, phase coexistence, long-range order, and giant number fluctuations, features typically associated with ordered phases of models where self-propelled particles with overdamped dynamics are considered.