scholarly journals The Riemann Problem and the Existence of Weak Solutions to a System of Mixed-Type in Dynamic Phase Transition

1998 ◽  
Vol 146 (2) ◽  
pp. 287-319 ◽  
Author(s):  
Harumi Hattori
Keyword(s):  
Phase Transition ◽  
Riemann Problem ◽  
Weak Solutions ◽  
Mixed Type ◽  
Existence Of Weak Solutions ◽  
Dynamic Phase ◽  
Dynamic Phase Transition

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

Mechanism of dynamic phase transition and synchronous stability in Hindmarsh–Rose neuronal network

2018 ◽  
Vol 32 (28) ◽  
pp. 1850308
Author(s):  
Shi-Dong Liang ◽  
Haoqi Li ◽  
Yuefan Deng
Keyword(s):  
Phase Transition ◽  
Parameter Space ◽  
Lyapunov Stability ◽  
Neuronal Network ◽  
Neuronal Model ◽  
Index Formula ◽  
Neuronal Dynamics ◽  
Dynamical Phase ◽  
Dynamic Phase ◽  
Dynamic Phase Transition

The neuronal dynamics plays an important role in understanding the neurological phenomena. We study the mechanism of the dynamic phase transition and its Lyapunov stability of a single Hindmarsh–Rose (HR) neuronal model. We propose an index [Formula: see text] to express the dynamical phase of the HR neurons. When [Formula: see text] the neuron is in the pure resting state, and when [Formula: see text] the neuron closes to the pure spiking phase, while when [Formula: see text] the neuron runs in the bursting phase. Based on this method, we investigate numerically the phase diagram of the HR neuronal model in the parameter space. We find that two mechanisms governed the HR neuronal dynamic phase transition, the phase transition and crossover transition in the different regions of the parameter space. Moreover, we analyze the equilibrium point stability of the HR neuronal model based on the Lyapunov stability method. We study the synchronous stability of the HR neuronal network based on the master stability function method and give the phase diagrams of the maximum Lyapunov exponents in the parameter space of networks. The regions of the synchronous stabilities in the parameter space depend on the couplings of the HR neurons of the membrane potential and the flux of the fast ion channel between the HR neurons. These results help to understand the HR neuronal dynamics and the synchronous stability of the HR neuronal networks.

scholarly journals NONEQUILIBRIUM PHASE TRANSITIONS IN MODEL FERROMAGNETS: A REVIEW

2005 ◽  
Vol 16 (11) ◽  
pp. 1631-1670 ◽  
Author(s):  
MUKTISH ACHARYYA
Keyword(s):  
Magnetic Field ◽  
Phase Transition ◽  
Transition Point ◽  
Time Dependent ◽  
Heisenberg Ferromagnet ◽  
Dynamic Transition ◽  
Dynamic Phase ◽  
Nonequilibrium Dynamic ◽  
Dynamic Phase Transition ◽  
Dynamic Transitions

The thermodynamical behaviors of ferromagnetic systems in equilibrium are well studied. However, the ferromagnetic systems far from equilibrium became an interesting field of research in last few decades. Recent exploration of ferromagnetic systems in the presence of a steady magnetic field are also studied by using standard tools of equilibrium statistical physics. The ferromagnet in the presence of time-dependent magnetic field, shows various interesting phenomena. An usual response of a ferromagnet in the presence of a sinusoidally oscillating magnetic field is the hysteresis. Apart from this hysteretic response, the nonequilibrium dynamic phase transition is also a very interesting phenomenon. In this chapter, the nonequilibrium dynamic phase transitions of the model ferromagnetic systems in presence of time-dependent magnetic field are discussed. For this kind of nonequilibrium phase transition, one cannot employ the standard techniques of equilibrium statistical mechanics. The recent developments in this direction are mainly based on numerical simulation (Monte Carlo). The Monte Carlo simulation of kinetic Ising model, in presence of sinusoidally oscillating (in time but uniform over space) magnetic field, is extensively performed to study the nonequilibrium dynamic phase transition. The temperature variations of dynamic order parameter, dynamic specific heat, dynamic relaxation time etc. near the transition point are discussed. The appearance and behaviors of a dynamic length scale and a dynamic time scale near the transition point are also discussed. All these studies indicate that this proposed dynamic transition is a nonequilibrium thermodynamic phase transition. The disorder (quenched) induced zero temperature (athermal) dynamic transition is studied in random field Ising ferromagnet. The dynamic transition in the Heisenberg ferromagnet is also studied. The nature of this transition in the Heisenberg ferromagnet depends on the anisotropy and the polarisation of the applied time varying magnetic field. The anisotropic Heisenberg ferromagnet in the presence of elliptically polarised magnetic field shows multiple dynamic transitions. This multiple dynamic transitions in anisotropic Heisenberg ferromagnet are discussed here. Recent experimental evidences of dynamic transitions are also discussed very briefly.

Photocontrol of a dynamic phase transition in the CDW state

2002 ◽  
Vol 12 (9) ◽  
pp. 297-301
Author(s):  
K. Miyano ◽  
N. Ogawa
Keyword(s):  
Phase Transition ◽  
Internal Stress ◽  
One Dimensional ◽  
Dynamic Phase ◽  
Wide Range ◽  
Dynamic Phase Transition

A strong photoinduced suppression of the creep to slide dynamic phase transition was discovered in a quasi one-dimensional CDW material, K0.3MoO3. It is manifested in hysteretic “photoresistivity", an increase in the resistance by orders of magnitude under the illumination of light of modest intensity. Based on a wide range of measurements, it is concluded that the effect is due to the release of the internal stress by photoexcited jumps of the CDW phase over pinning sites: a phenomenon which can be effective only in a system, in which the thermally excited quasiparticles are negligibly small and hence the `bare' CDW properties emerge.

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