GSM:020117Landau f=a(T-Tc)m^2/2+b m^4/4-hm: m±=±1/2, Delta f=-1/8, chi-=1/4 at T=3; chi+=1/2 at T=5; beta=1/2,gamma=1,delta=3; Delta C=a^2 Tc/(2b)Gsm2 Landau Mean Field MinimaGsm2 Landau Susceptibility Heat Capacity JumpPrivateLabels unavailablePrivate saving unavailableCritical phenomena, scaling, and renormalization groupGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:0201182D Ising FSS: beta=1/8,gamma=7/4,nu=1; t=0 m~L^{-1/8}, chi~L^{7/4}; L->2L ratios 2^{-1/8} and 2^{7/4}; shift halves; 2beta+gamma=2nu, alpha=0, eta=1/4Gsm2 2D Ising Finite Size ScalingGsm2 Hyperscaling Fisher RushbrookePrivateLabels unavailablePrivate saving unavailableCritical phenomena, scaling, and renormalization groupGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020119Kibble-Zurek: nu=1,z=2,d=1,tauQ/tau0=1000 give |tHat|=100 tau0, |epsHat|=0.1, xiHat=10 xi0, n=0.1/xi0Gsm2 Kibble Zurek Freezeout TimeGsm2 Kibble Zurek Defect DensityPrivateLabels unavailablePrivate saving unavailableCritical phenomena, scaling, and renormalization groupGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020220Onsager L=[[2,1],[1,3]], X=(1,-1/2): J=(3/2,-1/2), sigma=7/4, eigenvalues (5±sqrt5)/2>0; J2=0 gives X2=-X1/3, J1=(5/3)X1, sigma=(5/3)X1^2Gsm2 Onsager Linear Response FluxesGsm2 Onsager Entropy Production SpectrumPrivateLabels unavailablePrivate saving unavailableNonequilibrium ensembles and entropy productionGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020221Two-state master k01=3,k10=1: p1dot=3-4p1, p1*=3/4, p1(t)=(3/4)(1-e^{-4t}), tau=1/4 s, DeltaE=-kBT ln3, Cov=3e^{-4t}/16Gsm2 Two State Master EquationGsm2 Stationary Occupation CorrelatorPrivateLabels unavailablePrivate saving unavailableNonequilibrium ensembles and entropy productionGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020322BGK f=fM+(f0-fM)e^{-t/tau}; matched moments conserved; Pi_xy(t)=Pi0 e^{-t/tau}; eta=p tau=0.4 Pa s; t=tau ln4 gives Pi=Pi0/4Gsm2 Bgk Relaxation SolutionGsm2 Bgk Newtonian Shear ViscosityPrivateLabels unavailablePrivate saving unavailableKinetic theory and the Boltzmann equationGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020323Maxwell speed s^2=kBT/m: f(v)=sqrt(2/pi) v^2 e^{-v^2/(2s^2)}/s^3; vmp=sqrt2 s, mean=2sqrt(2/pi)s, vrms=sqrt3 s; P(v<vmp)=erf(1)-2/(e sqrtpi)~0.427593; s=20 m/s gives 20sqrt2, 40sqrt(2/pi), 20sqrt3Gsm2 Maxwell Speed MomentsGsm2 Maxwell Speed Cdf TO ModePrivateLabels unavailablePrivate saving unavailableKinetic theory and the Boltzmann equationGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020424OU dx=-gamma x dt+sqrt(2D)dW: mean x0 e^{-gamma t}, Var=(D/gamma)(1-e^{-2 gamma t}); stationary D/gamma, C(Delta)=(D/gamma)e^{-gamma|Delta|}, S(omega)=2D/(gamma^2+omega^2); gamma=2,D=8,x0=3,t=ln2/2 => mean 3/2, var 3, C(1)=4e^{-2}, S(2)=2Gsm2 OU Transient MomentsGsm2 OU Stationary SpectrumPrivateLabels unavailablePrivate saving unavailableMaster equations, Langevin equations, and Fokker-Planck theoryGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020425Overdamped Kramers kesc=sqrt[U''(xa)|U''(xb)|] e^{-beta DeltaU}/(2 pi zeta); curvatures 4,1, zeta=2, beta DeltaU=5 => k=e^{-5}/(2pi) s^{-1}, MFPT=2 pi e^5 s; lower barrier by kBT multiplies rate by e; double T at fixed DeltaU gives ratio e^{2.5}Gsm2 Kramers Overdamped RateGsm2 Kramers Temperature ScalingPrivateLabels unavailablePrivate saving unavailableMaster equations, Langevin equations, and Fokker-Planck theoryGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020426Langevin m vdot=-zeta v-k x+xi, <xi xi>=2 zeta kBT delta; eq <x^2>=kBT/k, <v^2>=kBT/m, <xv>=0, <E>=kBT; Cx(t)=(kBT/k) e^{-gamma t}[cos omega_d t+(gamma/omega_d)sin omega_d t]; m=1,k=4,zeta=2 => gamma=1, omega_d=sqrt3, Cx(pi/sqrt3)/Cx0=-e^{-pi/sqrt3}, Sx(2)=kBT/4Gsm2 Langevin Harmonic EquipartitionGsm2 Langevin Harmonic Position SpectrumPrivateLabels unavailablePrivate saving unavailableMaster equations, Langevin equations, and Fokker-Planck theoryGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020527Debye tau Mdot+M=chi0 h, e^{-i omega t}: R=chi0 e^{-t/tau} Theta(t)/tau, step chi0 h0(1-e^{-t/tau}), chi=chi0/(1-i omega tau), x=1 both chi0/2 modulus chi0/sqrt2, int R=chi0, KK chi0=(2/pi)int_0^inf chi''/omega d omegaView access optionsGsm2 Debye Causal Impulse StepGsm2 Debye Fourier KK ON ShellPrivateLabels unavailablePrivate saving unavailableLinear response, Green-Kubo relations, and transportGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020528Drude m vdot+m v/tau=qE, e^{-i omega t}: sigma=sigma0/(1-i omega tau), sigma0=n q^2 tau/m, x=1 Re=Im=sigma0/2 modulus sigma0/sqrt2; nq^2/m=4,tau=1/2,omega=2,E0=3 give sigma0=2,sigma=1+i,modulus sqrt2,power 9/2, one-sided sum 2piView access optionsGsm2 Drude AC Sigma MobilityGsm2 Drude Power Onesided F SumPrivateLabels unavailablePrivate saving unavailableLinear response, Green-Kubo relations, and transportGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020629Iid Exp(rate 1) sample mean: M(k)=1/(1-k) for k<1, lambda=-ln(1-k), lambda'(0)=lambda''(0)=1, I(x)=x-1-ln x (x>0), k*=1-1/x; x=2 gives k*=1/2, I=1-ln2, P~exp[-N(1-ln2)]; I(1)=0, I''(1)=1, I->inf at 0+ and +infView access optionsGsm2 Exponential Scgf CramerGsm2 Exponential Sample Mean SaddlePrivateLabels unavailablePrivate saving unavailableFluctuation theorems and large-deviation structureGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020630Independent Poisson rates 4 and 1: lambda=4(e^k-1)+(e^{-k}-1), mean 3, var 5, y=(j+sqrt(j^2+16))/8, I=j ln y-4(y-1)-(y^{-1}-1), I(0)=1, A=ln4, GC and I(j)-I(-j)=-A j, Sigma=3 ln4, TUR 5 ln4/3>2View access optionsGsm2 Biased Poisson Scgf SaddleGsm2 Gallavotti Cohen Tur ProductPrivateLabels unavailablePrivate saving unavailableFluctuation theorems and large-deviation structureGraduate Statistical Mechanics II · Mixed reviewProgress not loaded
GSM:020631N unbiased +/-1 spins: lambda=ln cosh k, mean 0, var 1, I(m)=m atanh m-ln cosh(atanh m)=1/2[(1+m)ln(1+m)+(1-m)ln(1-m)]; m=1/2 gives k*=(1/2)ln3, I=3ln(3/2)/4+ln(1/2)/4~0.130812; I(0)=0, I(+/-1)=ln2, small m I=m^2/2+O(m^4)View access optionsGsm2 Spin Magnetization Scgf RateGsm2 Spin Ldp Saddle EndpointsPrivateLabels unavailablePrivate saving unavailableFluctuation theorems and large-deviation structureGraduate Statistical Mechanics II · Mixed reviewProgress not loaded