Equilibrium, out of equilibrium and consequences

Transcription

1 Equilibrium, out of equilibrium and consequences Katarzyna Sznajd-Weron Institute of Physics Wroc law University of Technology, Poland SF-MTPT Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 1 / 63

2 Agenda 1 Equilibrium and H-Boltzmann theorem 2 Phase transitions in 1D 3 Example: Zero temperature Glauber dynamics in 1D 4 Markov chains and Monte Carlo simulations 5 Spin dynamics Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 2 / 63

3 H-Boltzmann theorem Microscopic reversibility P r (t) - probability of finding a system in the state r (one of possible states) at time t A system is certainly in one of possible states, i.e. r P r(t) = 1 W r s W rs - transition probability per unit time from state r to s Evolution equation (balance): dp r dt = P s W s r P r W r s s r s r }{{}}{{} s r r s If there is a symmetry W sr = W rs (microscopic reversibility): (1) dp r dt = s r W rs (P s P r ) (2) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 3 / 63

4 What is H? Let s define H as: H-Boltzmann theorem H < ln P r > r P r ln P r, (3) therefore: dh dt = d dt = r = r ( ) P r ln P r r ( ln P r dp r ) dt + P 1 dp r r P r dt ) ( dp r ln P r dt + dp r dt = r dp r dt (ln P r + 1) (4) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 4 / 63

5 Evolution of H H-Boltzmann theorem dh dt = r dp r dt (ln P r + 1), (5) but: dp r dt = s W rs (P s P r ), (6) and finally: dh dt = r W rs (P s P r ) (ln P r + 1), (7) s Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 5 / 63

6 H-Boltzmann theorem On the other hand we can replace r by s: H = P s ln P s (8) s and we obtain: dh dt = s Recalling that also: dh dt = r we can write: dh dt W sr (P r P s ) (ln P s + 1). (9) r W rs (P s P r ) (ln P r + 1), (10) s = 1 W rs (P s P r ) (ln P r + 1) 2 r s + 1 W sr (P r P s ) (ln P s + 1) (11) 2 s r Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 6 / 63

7 H-Boltzmann theorem Use the symmetry... dh dt = 1 W rs (P s P r ) (ln P r + 1) 2 r s + 1 W sr (P r P s ) (ln P s + 1) (12) 2 s r We use the microscopic reversibility (symmetry) condition W rs = W sr : dh = 1 W rs (P r P s ) (ln P r + 1) dt 2 r s + 1 W rs (P r P s ) (ln P s + 1) 2 s r = 1 W rs (P r P s ) (ln P r ln P s ). (13) 2 r s Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 7 / 63

8 H-Boltzmann theorem Evolution of H and symmetry dh dt = 1 W rs (P r P s ) (ln P r ln P s ). (14) 2 r s Note that (P r P s ) (ln P r ln P s ) 0, because: P r > P s then ln P r > ln P s (P r P s ) (ln P r ln P s ) > 0, P r < P s then ln P r < ln P s (P r P s ) (ln P r ln P s ) > 0, P r = P s then ln P r = ln P s (P r P s ) (ln P r ln P s ) = 0. Moreover W rs 0 (probability), therefore: dh dt 0 (= for P s = P r ) (15) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 8 / 63

9 H-Boltzmann theorem H-Boltzmann theorem dh dt 0 (= for P s = P r ) (16) H < ln P r > decreases in time until P s = P r (equal a priori probabilities - equilibrium) What is H? hence: S k B < ln P r > S = k B H, (17) dh dt 0 (= for P s = P r ) ds dt 0 (= for P s = P r ) (18) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 9 / 63

10 H-Boltzmann theorem H-Boltzmann theorem dh ds 0 (= for Ps = Pr ) 0 (= for Ps = Pr ) dt dt Microscopic interpretation of a second law of thermodynamics Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT (19) 10 / 63

11 H-Boltzmann theorem Comments on H-Boltzmann theorem Valid for isolated systems: dp r dt = P s W sr P r W rs +? (20) s s }{{}}{{} s r r s Microscopic reversibility is needed: W rs = W sr Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 11 / 63

12 Detailed balance Master equation and detailed balance P r (t) - probability of finding a system in the state r (one of possible states) at time t A system is certainly in one of possible states, i.e. r P r(t) = 1 W r s W rs - transition probability per unit time from state r to s Master equation: Detailed balance: dp r dt = P s W s r P r W r s s r s r }{{}}{{} s r r s (21) P s W sr = P r W rs dp r dt = 0 (22) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 12 / 63

13 Master equation and detailed balance Steady state and stationarity It is possible that detailed balance is not fulfilled but: dp r = 0 (23) dt Condition for a stationarity (steady state): P s W s r = P r W r s (24) s r s r }{{}}{{} s r r s Steady state in and out of equilibrium: Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 13 / 63

14 Questions: Master equation and detailed balance Consider a system with 3 possible states A, B, C. Transition probabilities w i j 0 for any i j (where i and j denotes one of possible states A, B, C). What are conditions for w i j, to fulfill detailed balance: What are stationarity conditions? P s W sr = P r W rs? (25) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 14 / 63

15 Phase transitions in 1D Diffusion on a circle 1D lattice with periodic boundary conditions Each site is occupied by a particle (A, B, C) C B A B B Random sequential updating: B A AB q BA, BC q CB, CA q AC, BA 1 AB, CB 1 BC, AC 1 CA. A C C B C The number of particles (N A, N B, N C ) is conserved [1] M. R. Evans, Y. Kafri, H. M. Koduvely and D. Mukamel, Phys. Rev. Lett. 80, 425 (1998) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 15 / 63

16 Phase transitions in 1D Phase separation For q = 1 disorder For q < 1 walls AB, BC, CA are stable Domains are created...aabbccaaab... B C B A B B A Domains grow in time ln t/ ln q Thermodynamic limit separtion into 3 domains A C C B C Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 16 / 63

17 Phase transitions in 1D Phase transitions in 1D? - Landau argument Equilibrium minimum of a free energy: F = E TS Change of a free energy (extensive): F = E T S = ɛ T S Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 17 / 63

18 What is entropy? Phase transitions in 1D S = k B ln W L {}}{ AAAABBBB }{{} L 1 Change of a free energy W = L 1 S = k B ln(l 1) F = ɛ T S = ɛ Tk B ln(l 1) In thermodynamic limit, i.e. for L F < 0 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 18 / 63

19 Phase transitions in 1D Landau assumptions Energy of a domain wall ɛ is finite F = ɛ Tk B ln(l 1) Short-range interactions no interactions between domain walls In 1D Ising model with J(r) r 1 σ there is an order in low temperatures T > 0 for σ 1 T > 0 otherwise entropy argument is not important F = E TS Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 19 / 63

20 Phase transitions in 1D Other argument - dynamics of a system AAAABBBBBAAA Transition disorder order annihilation of domain walls Domain ended with two walls costs 2ɛ In 1D there is no energy profit related to shrinking of a domain There is no force that eliminates domain walls (bonds) in 1D Assumption: no long-range interactions between bonds Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 20 / 63

21 Phase transitions in 1D Other suggestions: 1 Exact result for 1D Ising model 2 Van Hove theorem: No phase transitions for in 1D system of hard cores with short-range interactions 3 Mermin-Wagner theorem: No phase transitions in 1D and 2D systems with continuous symmetry for short-range interactions There is no general theorem for non-existence of a phase transitions in 1D! Literature: M. R. Evans, Braz. J. Phys. 30, 42 (2000) J. A. Cuesta, A. Sanchez, J. Stat. Phys. 115, 869 (2004) N. Theodorakopoulos, Physica D 216, 185 (2006) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 21 / 63

22 Phase transitions in 1D L. van Hove, Physica 16 (1950) 137 Free energy has been calculated for a hard-core model with short-range interactions Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 22 / 63

23 Phase transitions in 1D Limitations Identical particles were considered (hard cores of radius d 0 ) Homogeneous system identical elements Integrations on a distance d d 1 Short-range interactions Interaction energy depended only on a distance between particles (and not positions) No heterogeneous field or interactions Transfer matrix method have been used only maximum eigenvalue in thermodynamic limit Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 23 / 63

24 Phase transitions in 1D Equilibrium phase transitions in 1D Heterogeneous systems Long-range interactions Infinite energy of interactions At temperature T = 0 Out of equilibrium? Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 24 / 63

25 Out of equilibrium Phase transitions in 1D H-Boltzmann theorem Isolated system W sr = W rs (micro rev) H < ln P r >= S/k B dh dt 0 (= dla P s = P r ) Detailed balance: P eq s W sr = Pr eq W rs Type I Hermitian Hamiltonian Steady states given by the Gibbs-Boltzmann dist Initial state far from equilibrium In thermodynamic limit equilibrium is never reached Type II No Hamiltonian Dynamics given by transition probabilities No detailed balance Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 25 / 63

26 Example: Zero temperature Glauber dynamics in 1D Ising model in 1D 1D lattice with periodic boundary conditions Each site occupied with σ i = ±1 Interactions described by: H = J L σ i σ i+1 i=1 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 26 / 63

27 Example: Zero temperature Glauber dynamics in 1D Generalized Glauber dynamics at T = 0 Roy J. Glauber, Journal of Mathematical Physics (1963) σ i (t) σ i (t + dt) with prob. 1 ifδe < 0, W (δe) = W 0 if δe = 0, 0 if δe > 0, σ i 1 σ i σ i+1 W W 0 W 0 W 0 W 0 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 27 / 63

28 Example: Zero temperature Glauber dynamics in 1D Updating 1 Sequential consider single spin in an elementary time step dt (Glauber, Metropolis) 2 Synchronous all spins are considered at the same time (as in cellular automata); for W 0 = 1 absorbing antiferromagnetic state: 3 c-synchronous partially synchronous K. Sznajd-Weron and S. Krupa, Phys. Rev. E 74, (2006) sequential }{{} c=1/l synchronous }{{} c=1 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 28 / 63

29 Example: Zero temperature Glauber dynamics in 1D Order parameter and phase diagram ρ = 1 L (1 σ i σ i+1 ) 2L i=1 <ρ st > W c Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 29 / 63

30 Example: Zero temperature Glauber dynamics in 1D Discontinuous phase transitions for c = 1 1 <ρ> W 0 =0.8 W =0.7 0 W 0 =0.6 W =0.5 0 W 0 =0.4 W 0 =0.3 W = t Monte Carlo simulations: time evolution of bonds for the systems size L = 160, averaging over 10 4 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 30 / 63

31 Example: Zero temperature Glauber dynamics in 1D Scaling for c = 1: β = 0, ν = 1 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 31 / 63

32 Example: Zero temperature Glauber dynamics in 1D Discontinuous phase transitions Jump of an order parameter ρ Trivial critical exponents β = 0, ν = 1 Phase coexistence for W 0 = 1 2 Hysteresis cycle metastable states, dependence on the initial conditions 1 1 <ρ st > 0.5 <ρ st > W 0 W 0 Figure : left: c = 1, right: c = 0.95 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 32 / 63

33 Katarzyna Sznajd-Weron Figure (WUT) : c = Equilibrium, 0.99, Wout c of = equilibrium 0.53, and β = , ν = 1 SF-MTPT 33 / 63 Example: Zero temperature Glauber dynamics in 1D Continuous phase transitions for c < 1 <ρ st > W 0 L β/ν <ρ st > 5 4 L=32 3 L=48 2 L=64 L=100 1 L=123 L= (W 0 W c )L 1/ν Figure : c = 0.9, W c = 0.6, β = 0.4, ν = <ρ st > W 0 L β/ν <ρ st > L=32 2 L=48 L=64 1 L=100 L=123 L= (W 0 W c )L 1/ν

34 Example: Zero temperature Glauber dynamics in 1D Critical exponents and the type of transition c W c β ν Discontinuous c = 1, Continuous(W 0 = 1) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 34 / 63

35 Monte Carlo and dynamics Monte Carlo and dynamics Monte Carlo is a method of calculating quantities that can be presented as the expected value of certain probabilistic distributions Let s m is our quantity and we know that m = EX of a certain random variable X If we are able to generate S 1, S 2,... that are iid (independent identically distributed) from distribution of X than the law of large numbers: 1 lim n n (S S n ) = m. (26) Monte Carlo estimate a quantity m by a properly chosen sample of n elements Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 35 / 63

36 Monte Carlo and dynamics How to choose a sample properly in Statistical Physics? System in a given external conditions equilibrium (or approaches equilibrium) Probability density function in equilibrium: P eq (α) p α = 1 Z exp( βe α), (27) where Z = α exp( βe α) partition function Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 36 / 63

37 Monte Carlo and dynamics Example: evolution of a system Consider Ising model Initial condition: m(0) = m 0 = 1 (low temperature state T 0, in general m = ±1). Initial distribution of our random variable m is given by P m (0) = δ m,m0 = δ m,1. What will distribution of m at temperature T = T 1 >> 0? From statistical physics system approaches equilibrium given by the Boltzmann-Gibbs distribution Evolution of m(t) and its distribution In the probability theory such an evolution stochastic processes Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 37 / 63

38 Monte Carlo and dynamics Stochastic processes Stochastic processes evolution of some random value X t, or system, over time t (function) From a statistical physics point of view ensemble of trajectories (realizations of a given process) What can be our random variable? Particularly convenient is a microstate (configuration) of a system. My recomendation: Ch. M. Grinstead and J. L. Snell Introduction to Probability Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 38 / 63

39 Monte Carlo and dynamics Markov process We have a set of states S = s 1, s 2,..., s r The process starts in one of these states and moves successively from one state to another If the chain is currently in state s i, then it moves to state s j at the next step with a probability W ij The probability W ij does not depend upon which states the chain was in before the current state R. A. Howard a frog jumping on a set of lily pads. The frog starts on one of the pads and then jumps from lily pad to lily pad with the appropriate transition probabilities Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 39 / 63

40 Monte Carlo and dynamics Example: Weather in the Land of Oz The Land of Oz is blessed by many things, but not by good weather. They never have two nice days in a row. If they have a nice day, they are just as likely to have snow as rain the next day. If they have snow or rain, they have an even chance of having the same the next day. If there is change from snow or rain, only half of the time is this a change to a nice day. Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 40 / 63

41 Monte Carlo and dynamics Example: Weather in the Land of Oz We form a Markov chain We take as states the kinds of weather: R (rain), N (nice), and S (snow). From the information we determine the transition probabilities (transition matrix): Check it! W = R N S R 1/2 1/4 1/4 N 1/2 0 1/2 S 1/4 1/4 1/2 (28) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 41 / 63

42 Monte Carlo and dynamics Markov chain Let s define Markov process with discrete time t 1 < t 2 <... < t n, for a finite number of states σ 1, σ 2, σ 3,..., σ N : Random variable X t describes a state of a system at t. Conditional probability that X tn = σ in : P(X tn = σ in X tn 1 = σ in 1, X tn 2 = σ in 2,..., X t1 = σ i1 ), (29) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 42 / 63

43 Markov chain Monte Carlo and dynamics Markov property: P(X tn = σ in X tn 1 = σ in 1,..., X t1 = σ i1 ) = P(X tn = σ in X tn 1 = σ in 1 ). (30) Above conditional probability transition probability from i to j therefore: W ij = W (σ i σ j ) = P(X tn = σ j X tn 1 = σ i ). (31) W ij 0, W ij = 1, (32) j Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 43 / 63

44 Monte Carlo and dynamics Master equation Probability that at time t n system will be in a state σ j P(X tn = σ j ) = P(X tn = σ j X tn 1 = σ i )P(X tn 1 = σ i ) (33) = W ij P(X tn 1 = σ i ). (34) Evolution described by the Master Equation: dp(σ j, t) dt = i W ji P(σ j, t) + i W ij P(σ i, t). (35) W ji P eq (σ j ) = W ij P eq (σ i ). (36) P eq (σ j ) = Z 1 e E(σ j )/T. (37) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 44 / 63

45 Spin dynamics Voter model Toy model Each site of a graph (or lattice) is occupied by a particle in one of q states In each elementary time step we choose one spin It takes a state of a randomly chosen nn Dynamics each finite system reaches of q absorbing states (all the same) Exit time (time to reach an absorbing state) depends on the system size, dimension and q Most common q = 2 (spin s = 1/2) s( x) = ±1 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 45 / 63

46 Spin dynamics Voter model - algorithm Set an initial conditions e.g. for each x = 1,..., N with probability p set s(x, 0) = 1, and with 1 p set s(x, 0) = 1 1 Choose randomly (uniform distribution) a spin s(x, t) 2 Choose randomly one of its nn s(y, t) 3 If s(x, t) s(y, t) then s(x, t + t) = s(y, t) 4 Increase time t t + t 5 Check if all spins are the same. If not goto 1 N elementary steps increase time by 1, i.e. t = 1/N Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 46 / 63

47 Spin dynamics Voter model - transition probability Transition rate w x ( s) spin s(x) flips: [ w x ( s) = 1 1 s(x) 2 z hence: s(x, t + t) = y <x> s(y) { s(x, t) with probability 1 wx ( s) t s(x, t) with probability w x ( s) t ] (38) (39) prawdopodobieństwo = 0 prawdopodobieństwo = 1/4 prawdopodobieństwo = 1/2 prawdopodobieństwo = 3/4 prawdopodobieństwo = 1 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 47 / 63

48 Spin dynamics Voter model - master equation (ME) State is fully described by P( s, t) probability that at time t a system has a state s = [s 1, s 2,..., s N ]. Number of states 2 N Probability distribution evolves due to ME dp( s) dt = x w x ( s)p( s) + x w x ( s x )P( s x ), (40) where s x differs from s by spin at site x. Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 48 / 63

49 Spin dynamics Voter model magnetization Spin at site x changes by 2s(x) with a rate w x ( s) w x, therefore average spin at site denoted by S(x) < s(x) >: Recalling that: ds(x) dt w x ( s) = 1 2 = 2 < s(x)w x > (41) [ 1 s(x) z y <x> s(y) ] We obtain (calculate this!): ds(x) dt = S(x) + 1 z y <x> S(y) (42) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 49 / 63

50 Spin dynamics Voter model magnetization ds(x) dt Average magnetization: = S(x) + 1 z y <x> S(y) (43) m x S(x) N (44) From (43) we obtain that on average: What are consequences? dm dt = 0 (45) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 50 / 63

51 Spin dynamics Voter model exit Probability Consider a finite system with the initial probability of +1 equal p, and 1 equal 1 p Therefore initial magnetization m 0 = 2p 1 Eventually system reaches absorbing state m = 1 with probability E(p) or m = 1 z 1 E(p) Final magnetization: m = E(p) 1 + [1 E(p)] ( 1) = 2E(p) 1 (46) Previously, we have obtained that m = m 0 = 2p 1 E(p) = p Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 51 / 63

52 Spin dynamics Glauber dynamics for an Ising model H = <i,j> J ij S i S j S i = ±1 (47) We choose in each time step single spin and S i S i with probability that depends on the energy change energia maleje energia rośnie brak zmiany energii Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 52 / 63

53 Spin dynamics Glauber dynamics - transition rates (TR) Detailed balance has to be fulfilled: P eq ( s)w i ( s) = P eq ( s i )w i ( s i ), (48) where s denotes one of 2 N possible states s i is created from s by flipping i-th spin Probability density at equilibrium: hence: P eq ( s) = exp[ βh( s)]. (49) Z w i ( s) w i ( s i ) = P eq( s i ) P eq ( s) = exp[ βs i j <i> J ijs j ] exp[βs i j <i> J ijs j ] (50) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 53 / 63

54 Spin dynamics Glauber dynamics - transition rates (TR) w i ( s) w i ( s i ) = P eq( s i ) P eq ( s) = exp[ βs i j <i> J ijs j ] exp[βs i j <i> J ijs j ] Use the relation for s ± 1 (show it!): =? (51) e As = cosh(a) + s sinh(a) = cosh(a)[1 + s tanh(a)]. (52) then: w i ( s) 1 s i tanh (β ) w i ( s i ) = j <i> J ijs j 1 + s i tanh (β ) (53) j <i> J ijs j Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 54 / 63

55 Spin dynamics Glauber dynamics - transition rates (TR) The simplest choice w i ( s) 1 s i tanh (β ) w i ( s i ) = j <i> J ijs j 1 + s i tanh (β ) (54) j <i> J ijs j w i ( s) = W 0 [1 s i tanh ( β j <i> J ij s j )], (55) where W 0 is a constant chosen by Glauber as 1/2. What is a physical meaning of W 0? Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 55 / 63

56 Spin dynamics Ising model - MFA Let s write the Hamiltonia H = <i,j> JS i S j H = i h i S i, h i = 1 2 Js j. (56) j <i> Local field (from other spins) acts on every spin MFA: h i = h = Jzm/2. As we have seen it can be obtained from < δs i δs j >= 0. (57) Because s i =< s i > +δs i we obtain: < s i s j >=< s i >< s j >, (58) from mathematical point of view s i and s j are independent. Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 56 / 63

57 Spin dynamics Ising model on a complete graph Every spin interact with any other: H = 1 S i S j. (59) N i<j then: w i ( s) = W 0 [1 s i tanh = W 0 [1 s i tanh ( β ( β N j <i> J ij s j )] )] s j j i = W 0 [1 s i tanh (βm)]. (60) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 57 / 63

58 Spin dynamics Glauber dynamics - transition rates (TR) We obtain: therefore: s i (t + t) = w i ( s) = W 0 [1 s i tanh (βm)] (61) { si (t) with probability 1 w i ( s) t s i (t) with probability w i ( s) t Which means that spin at site i changes by 2s i with rate w i ( s) w i, therefore on average S i < s i >: (62) ds i dt = 2 < s iw i >= S i + tanh(βm). (63) Sum over all i: dm = m + tanh(βm). (64) dt Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 58 / 63

59 Spin dynamics Critical point and scaling At equilibrium: dm dt = 0 m = tanh(βm). (65) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 59 / 63

60 Spin dynamics Critical point and scaling At equilibrium: dm dt = 0 m = tanh(βm). (66) For β β c = 1 the only solution is m = 0. For β > β c = 1 there are two nontrivial solutions m ± Expand the equation above a critical point in βm: dm dt = tanh(βm) = (β c β)m 1 3 (βm) (67) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 60 / 63

61 Dynamics Spin dynamics dm dt = tanh(βm) = (β c β)m 1 3 (βm) (68) for T we have dm dt = (β c β)m m e t/τ, τ = (β c β) 1 (69) for T = T c : dm dt = 1 3 (βm) m t 1/2 (70) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 61 / 63

62 Spin dynamics Glauber dynamics in 1D We know that: in 1D simply: w i ( s) = W 0 [1 s i tanh w i ( s) = W 0 [1 s i tanh But foe ɛ = 0, ±1 we have (show it!): Hence: ( β j <i> J ij s j )], (71) ( βj s )] i 1 + s i+1. (72) 2 tanh(ɛx) = ɛ tanh(x). (73) [ ] s i 1 + s i+1 w i ( s) = W 0 1 γs i, γ = tanh(βj). (74) 2 Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 62 / 63

63 Spin dynamics Relaxation in 1D Glauber-Ising model As previously: ds i dt = 2 < s iw i >= S i + γ 2 (S i 1 + S i+1 ). (75) This equation can be solved assuming certain initial conditions (give an example) m(t) = m(0)e (1 γ)t γ = tanh(βj) (76) Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium and... SF-MTPT 63 / 63