The phase space of the focused cubic Schroedinger equation: A numerical study
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
In a paper of 1988 [41] on statistical mechanics of the nonlinear Schroedinger equation, it was observed that a Gibbs canonical ensemble associated with the nonlinear Schroedinger equation exhibits behavior reminiscent of a phase transition in classical statistical mechanics. The existence of a phase transition in the canonical ensemble of the nonlinear Schroedinger equation would be very interesting and would have important implications for the role of this equation in modeling physical phenomena; it would also have an important bearing on the theory of weak solutions of nonlinear wave equations. The cubic Schroedinger equation, as will be shown later, is equivalent to the self-induction approximation for vortices, which is a widely used equation of motion for a thin vortex filament in classical and superfluid mechanics. The existence of a phase transition in such a system would be very interesting and actually very surprising for the following reasons: in classical fluid mechanics it is believed that the turbulent regime is dominated by strong vortex stretching, while the vortex system described by the cubic Schroedinger equation does not allow for stretching. In superfluid mechanics the self-induction approximation and its modifications have been used to describe the motion of thin superfluid vortices, which exhibit a phase transition; however, more recently some authors concluded that these equations do not adequately describe superfluid turbulence, and the absence of a phase transition in the cubic Schroedinger equation would strengthen their argument. The self-induction approximation for vortices takes into account only very localized interactions, and the existence of a phase transition in such a simplified system would be very unexpected. In this thesis the authors present a numerical study of the phase transition type phenomena observed in [41]; in particular, they find that these phenomena are strongly related to the splitting of the phase space into distinctly different components. They point out the interesting fact that the phase space into distinctly different components. They point out the interesting fact that the phase transition type behavior of the discretized cubic Schroedinger equation can be observed in a discretization with as few as 2 points. The refinement of the discretization does not change the global picture qualitatively. The authors vary two parameters in the canonical ensemble of the cubic Schroedinger equation: the first parameter is the temperature, the second one is a certain constraint on the function space. They demonstrate that at a fixed low temperature, as the constraint varies, the canonical ensemble of the cubic Schroedinger equation undergoes a bifurcation which is manifested both in the change in the shape of the typical function and in a corresponding change of the structure of the phase space.
- Research Organization:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC); National Science Foundation (NSF)
- DOE Contract Number:
- AC03-76SF00098
- OSTI ID:
- 666170
- Report Number(s):
- LBNL-41872; ON: DE98058254; CNN: Grant DMS94-14631; Grant DMS89-19074; TRN: AHC29819%%326
- Resource Relation:
- Other Information: DN: Thesis submitted to the Univ. of California, Berkeley, CA (US); TH: Thesis (Ph.D.); PBD: May 1998
- Country of Publication:
- United States
- Language:
- English
Similar Records
Percolation in a self-avoiding vortex gas model of the {lambda} transition in three dimensions
Phase transitions and connectivity in three-dimensional vortex equilibria