We propose that the thermodynamics and the kinetics of the phase transition between wormhole and two black hole described by the two coupled SYK model can be investigated in terms of the stochastic dynamics on the underlying free energy landscape. We assume that the phase transition is a stochastic process under the thermal fluctuations. By quantifying the underlying free energy landscape, we study the phase diagram, the kinetic time and its fluctuations in details, which reveal the underlying thermodynamics and kinetics. It is shown that the first order phase transition between wormhole and two black hole described by two coupled SYK model is analogous to the Van der Waals phase transition. Therefore, the emergence of wormhole and two black hole phases, the phase transition and associated kinetics can be quantitatively addressed in our free energy landscape and kinetic framework through the dependence on the barrier height and the temperature.
Our work underscores not only the potential importance of zonal flows in other transitional turbulence situations9,10, but also shows the utility of coarse-grained effective models for non-equilibrium phase transitions, even to states as perplexing as fluid turbulence.
goldenfeld phase transitions djvu free
A good reference to applications is LatticeGas Cellular Automata: Simple Models of Complex Hydrodynamics by D. Rothman and S.Zaleski. These same authors have a somewhat different review article entitled Lattice-gasmodels of phase separation: interfaces, phase transitions, and multiphase flow inReviews of Modern Physics, 66, 1417 (1994).
The lecture course provides an introduction to the theory of classical and quantum phase transitions, to position-space as well as Wilson renormalisation-group theory. Emphasis will be set on broadly used spin models as well as bosonic field theories relevant in particular for applications in the field of ultracold atomic gases. Methodologically, the lecture will build on the basics of the operator as well as the path-integral approach to quantum field theory. Basic knowledge of quantum mechanics, statistical mechanics, and quantum field theory is presumed.
Content: Introduction- Classical phase transitions - phase diagram of water - Ehrenfest classification - continuous phase transitions - quantum phase transitions
Phase transition in the classical Ising model- Ising Hamiltonian - Spontaneous symmetry breaking - Thermodynamic properties - Phase transitions in the Ising model - Landau mean-field theory - Mean-field critical exponents - Correlation functions - Hubbard Stratonovich transformation - Functional-integral representation - Ginzburg-Landau-Wilson functional - Saddlepoint approximation and Gaussian effective action - Ginzburg criterion
Renormalisation-group theory in position space- Block-spin transformation - Transfer-matrix solution of the 1D Ising chain - RG stepping for the 1D and 2D Ising models - Critical point - RG fixed points - Relevant and irrelevant couplings - Universality and universality class - Renormalisation-group flows - Scaling properties of the free energy and of the two-point correlation function - Scaling relations between critical exponents - The scaling hypothesis
Wilson's Renormalisation Group - Perturbation theory - Linked-Cluster and Wick's theorems - Dyson equation - One-loop critical properties - Dimensional analysis - Momentum-scale RG - Gaussian fixed point - Wilson-Fisher fixed point - Epsilon-expansion - Critical exponents - Wave function renormalisation and anomalous dimension - Suppl. Mat.: Asymptotic expansions
Quantum phase transitions- Quantum Ising model - Mapping of the classical Ising chain to a quantum spin model - Universal scaling behaviour - Thermal as time-ordered correlators - Quantum to classical mapping - Perturbative spectrum of the transverse-field Ising model - Jordan Wigner transformation and exact spectrum - Universal crossover functions near the quantum critical point - Anomalous scaling dimension - Low-temperature and quantum critical regimes - Conformal mapping - Spectral properties close to criticality - Structure factor, susceptibility, and linear response - Relaxational response in the quantum critical regime
An analytic function can be Taylor expanded around the critical point. In a phase transition the derivatives of the free energy are discontinous, so the function can't be Taylor expanded. Hence free energy must be non-analytic.
About your second question, I think you are confusing things a bit: the transition is second order because the magnetization, which is the first derivative of the free energy with respect to the external magnetic field, is continuous at the critical point. However, susceptibility, which is the second derivative, is not continuous, meaning it is a second order phase transition.
The Wikipedia page is being sloppy. They mean that the free energy density is an analytic function of the mean-field order parameter, whereas at a thermal phase transition the free energy density is a non-analytic function of the temperature (or for a zero-temperature phase transition, of the external parameter being tuned across the transition).
I'll provide one possible answer to your question. I use the language of the Ising ferromagnet/paramagnet phase transition, but this is just for convenience as the same holds much more generally. Assume that the system is below the Curie temperature. Then, the free energy is an analytic function of the magnetic field both when $h>0$ and when $h
Namely, the real point is whether the free energy can be analytically continued from $h0$. It can be shown that the free energy possesses directional derivatives of all orders at $h=0$, that is,$$\frac\rm d^n\rm d_-h^n f(\beta,h)\vert_h=0$$exists for all $n$, where $\rm d/\rm d_-h$ means the left-derivative. Okay, so the question reduces to whether the Taylor series converges in a small disk around $0$. This is actually not the case, at least for finite-range models at low enough temperatures: there is no analytic continuation of the free energy beyond the transition point. This result was originally proved by Isakov in this paper. His analysis was restricted to the Ising model, but was more recently extended to a large class of 2-phases models; see the discussion in this paper. 2ff7e9595c
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