arXiv subjects
Publications and source records attributed to Jayanta K. Bhattacharjee.
We have studied the ground state of the Gross-Pitaevskii equation (nonlinear Schrodinger equation) for a Morse potential via a variational approach. It is seen that the ground state ceases to be bound when the coupling constant of the nonlinear term reaches a critical value. The disappearence of the ground state resembles a saddle node bifurcation.
Hopf bifurcations have been studied perturbatively under two broad headings, viz., super-critical and sub-critical. The criteria for occurrences of such bifurcations have been investigated using the renormalization group. The procedure has been described in details for both two and three dimensions and has been applied to several important models, including those by Lorenz and Rossler.
Starting from coupled Gross-Pitaevskii equation (GPE) describing the dynamics of binary Bose Einstein Condensate (BEC), we have explored the dynamics of the width of initial wave-packets by following two different approaches. We have followed a recent coherent state based approach and have shown that in binary BEC the oscillation of the width of the wave packet can give rise to instability at a certain condition. Alternately we have proceeded by considering the trial ground state solutions for both the species and through the Ehrenfest theorem shown the existence of an instability which may lead to phase separation of the two species.
We study the phase diagram of spatially inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) superconducting state using the Ginzburg-Landau(GL) free energy, derived from the microscopic Hamiltonian of the system, and notice that it has a very clear Lifshitz tricritical point. We find the specific heat jumps abruptly near the first-order line in the emergent phase diagram which is very similar to the recent experimental observation in layered organic superconductor. Comparison with experimental data allows us to obtain quantitative relations between the parameters of phenomenological free energy. The region of the phase diagram where the specific heat jumps can be probed by doing a dynamical analysis of the free energy.
Superconductor with a spatially-modulated order parameter is known as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductor. Using the time-dependent Ginzburg-Landau (TDGL) formalism we have theoretically studied the temporal behaviour of the equal-time correlation function, or the structure factor, of a FFLO superconductor following a sudden quench from the unpaired, or normal, state to the FFLO state. We find that quenching into the ordered FFLO phase can reveal the existence of a line in the mean-field phase diagram which cannot be accessed by static properties.
We show that the rare events present in dissipated work that enters Jarzynski equality, when mapped appropriately to the phenomenon of large deviations found in a biased coin toss, are enough to yield a quantitative work probability distribution for Jarzynski equality. This allows us to propose a recipe for constructing work probability distribution independent of the details of any relevant system. The underlying framework, developed herein, is expected to be of use in modelling other physical phenomena where rare events play an important role.
In a novel approach to studying viscous accretion flows, viscosity has been introduced as a perturbative effect, involving a first-order correction in the $α$-viscosity parameter. This method reduces the problem of solving a second-order nonlinear differential equation (Navier-Stokes equation) to that of an effective first-order equation. Viscosity breaks down the invariance of the equilibrium conditions for stationary inflow and outflow solutions, and distinguishes accretion from wind. Under a dynamical systems classification, the only feasible critical points of this "quasi-viscous" flow are saddle points and spirals. A linearised and radially propagating time-dependent perturbation gives rise to secular instability on large spatial scales of the disc. Further, on these same length scales, the velocity evolution equation of the quasi-viscous flow has been transformed to bear a formal closeness with Schrödinger's equation with a repulsive potential. Compatible with the transport of angular momentum to the outer regions of the disc, a viscosity-limited length scale has been defined for the full spatial extent over which the accretion process would be viable.
We show that the delta function potential can be exploited along with perturbation theory to yield the result of certain infinite series. The idea is that any exactly soluble potential if coupled with a delta function potential remains exactly soluble. We use the strength of the delta function as an expansion parameter and express the second-order energy shift as an infinite sum in perturbation theory. The analytical solution is used to determine the second-order energy shift and hence the sum of an infinite series. By an appropriate choice of the unperturbed system, we can show the importance of the continuum in the energy shift of bound states.
A first-order correction in the $α$-viscosity parameter of Shakura and Sunyaev has been introduced in the standard inviscid and thin accretion disc. A linearised time-dependent perturbative study of the stationary solutions of this "quasi-viscous" disc leads to the development of a secular instability on large spatial scales. This qualitative feature is equally manifest for two different types of perturbative treatment -- a standing wave on subsonic scales, as well as a radially propagating wave. Stability of the flow is restored when viscosity disappears.
The stationary background flow in the spherically symmetric infall of a compressible fluid, coupled to the space-time defined by the static Schwarzschild metric, has been subjected to linearized perturbations. The perturbative procedure is based on the continuity condition and it shows that the coupling of the flow with the geometry of space-time brings about greater stability for the flow, to the extent that the amplitude of the perturbation, treated as a standing wave, decays in time, as opposed to the amplitude remaining constant in the Newtonian limit. In qualitative terms this situation simulates the effect of a dissipative mechanism in the classical Bondi accretion flow, defined in the Newtonian construct of space and time. As a result of this approach it becomes impossible to define an acoustic metric for a conserved spherically symmetric flow, described within the framework of Schwarzschild geometry. In keeping with this view, the perturbation, considered separately as a high-frequency travelling wave, also has its amplitude reduced.
Transonicity in a spherically symmetric accreting system has been considered in both the stationary and the dynamic regimes. The stationary flow, set up as a dynamical system, has been shown to be greatly unstable to even the minutest possible deviation in the boundary condition for transonicity. With the help of a simple analytical model, and some numerical modelling, it has then been argued that the flow indeed becomes transonic and stable, when the evolution of the flow is followed through time. The time-dependent approach also shows that there is a remarkable closeness between an equation of motion for a perturbation in the flow, and the metric of an analog acoustic black hole.
For inviscid, rotational accretion flows driven by a general pseudo-Newtonian potential on to a Schwarzschild black hole, the only possible fixed points are saddle points and centre-type points. For the specific choice of the Newtonian potential, the flow has only two critical points, of which the outer one is a saddle point while the inner one is a centre-type point. A restrictive upper bound is imposed on the admissible range of values of the angular momentum of sub-Keplerian flows through a saddle point. These flows are very unstable to any deviation from a necessarily precise boundary condition. The difficulties against the physical realisability of a solution passing through the saddle point have been addressed through a temporal evolution of the flow, which gives a non-perturbative mechanism for selecting a transonic solution passing through the saddle point. An equation of motion for a real-time perturbation about the stationary flows reveals a very close correspondence with the metric of an acoustic black hole, which is also an indication of the primacy of transonicity.
In the presence of viscosity the hydraulic jump in one dimension is seen to be a first-order transition. A scaling relation for the position of the jump has been determined by applying an averaging technique on the stationary hydrodynamic equations. This gives a linear height profile before the jump, as well as a clear dependence of the magnitude of the jump on the outer boundary condition. The importance of viscosity in the jump formation has been convincingly established, and its physical basis has been understood by a time-dependent analysis of the flow equations. In doing so, a very close correspondence has been revealed between a perturbation equation for the flow rate and the metric of an acoustic white hole. We finally provide experimental support for our heuristically developed theory.
The stochastic quantisation technique of Parisi and Wu is extended to study non-equilibrium statistical mechanics. We show that this scheme is capable of handling white as well as coloured noises. PACS numbers: 64.60.-i; 64.60.Ak; 64.60.Fr; 64.60.Ht
The frequency and shear dependent critical viscosity at a correlation length $ξ=κ^{-1}$, has the form $η=η_{0}κ^{-x_η}G(z_{1},z_{2})$, where $z_{1}$ and $z_{2}$ are the independent dimensionless numbers in the problem defined as $z_{1}=\frac{-iω}{2Γ_{0}κ^{3}}$ and $z_{2}=\frac{-iω}{2Γ_{0}κ_{c}^{3}}$. The decay rate of critical fluctuations of correlation length $κ^{-1}$ is $Γ_{0}κ^{3}$ and $k_{c}$ is the effective wave number for which $Γ_{0}k_{c}^{3}=S$, the shear rate. The function $G(z_{1},z_{2})$ is calculated in a one loop self-consistent theory.
The long wavelength diffusion coefficient of a critical fluid confined between two parallel plates separated by a distance L is strongly affected by the finite size. Finite size scaling leads us to expect that the vanishing of the diffusion coefficient as ξ^{-1} for ξ< >L. We show that this is not strictlytrue. There is a logarthmic scaling violation. We construct a Kawasaki like scaling function that connects the thermodynamic regime to the extreme critical (ξ>>L) regime.
We arrange the loopwise perturbation theory for the critical viscosity exponent $x_η$, which happens to be very small, as a power series in $x_η$ itself and argue that the effect of loops beyond two is negligible. We claim that the critical viscosity exponent should be very closely approximated by $x_η=\frac{8}{15 π^2}(1+\frac{8}{3 π^2})\simeq 0.0685$.
We find that studying the simplest of the coupled non-equilibrium growth equations of Barabasi by self-consistent mode coupling requires the use of dressed vertices. Using the vertex renormalization, we find a roughness exponent which already in the leading order is quite close to the numerical value.