Thermodynamics and Dynamics of Systems with Long Range Interactions

1. Thermodynamics and dynamics of systems with long range interactions David Mukamel

2. Systems with long range interactions two-body interaction v(r) a 1/rs at large r with s<d, d dimensions examples: dipolar interactions, self gravitating systems, coulomb interactions, vortices in two dimensions, electrons interacting with laser field etc.

3. Thermodynamics: since the entropy may be neglected in the Thermodynamic limit the equilibrium state is just the ground state

4. In order to make the energy and the entropy scale in the same way one may rescale the Hamiltonian (Kac prescription) This is equivalent to taking the thermodynamic limit together with the limit as since self gravitating systems ferromagnetic dipolar systems

5. since In gravitational systems this would mean Is this limit realized in self gravitating systems?

6. Globular clusters are gravitationally bound concentrations of approximately ten thousand to one million stars, spread over a volume of several tens to about 200 light years in diameter.

7. For the M2 cluster N=150,000 stars R= 175 light years

8. Ferromagnetic dipolar systems v ~ 1/r3 D is the shape dependent demagnetization factor

9. Breaking of ergodicity in microcanonical ensemble Slow dynamics, diverging relaxation time Discuss features which result from the non-additivity Thermodynamics Negative specific heat in microcanonical ensemble Inequivalence of microcanonical (MCE) and canonical (CE) ensembles Dynamics

10. S Some general considerations Negative specific heat in microcanonical ensemble of non-additive systems. Antonov (1962); Lynden-Bell & Wood (1968); Thirring (1970) coexistence region in systems with short range interactions E0 = xE1 +(1-x)E2 S0 = xS1 +(1-x)S2 hence S is concave and the microcanonical specific heat is non-negative

11. On the other hand in systems with long range interactions (non-additive), in the region E1<E<E2 E0 = xE1 +(1-x)E2 S0 xS1 +(1-x)S2 S The entropy may thus follow the homogeneous system curve, the entropy is not concave. and the microcanonical specific heat becomes negative. compared with canonical ensemble where

12. _ + To study this question of ensembles’ inequivalence in more detail it is instructive to consider mean-field type interactions, e.g. in Ising models Although the canonical thermodynamic functions (free energy, entropy etc) are extensive, the system is non-additive

13. Ising model with long and short range interactions. d=1 dimensional geometry, ferromagnetic long range interaction J>0 The model has been analyzed within the canonical ensemble Nagel (1970), Kardar (1983)

14. Ground state + - + - + - + - + + + + + -1/2 K/J T/J 2nd order 1st order -1/2 0 K/J

15. Canonical (T,K) phase diagram

16. U/2 (+) segments U/2 (-) segments Microcanonical analysis Mukamel, Ruffo, Schreiber (2005); Barre, Mukamel, Ruffo (2001) U = number of broken bonds in a configuration Number of microstates:

17. Maximize to get s=S/N , =E/N , m=M/N where

18. s m continuous transition: discontinuous transition: In a 1st order transition there is a discontinuity in T, and thus there is a T region which is not accessible.

20. Microcanonical phase diagram

21. canonical microcanonical The two phase diagrams differ in the 1st order region of the canonical diagram

24. S In general it is expected that whenever the canonical transition is first order the microcanonical and canonical ensembles differ from each other.

25. Dynamics Microcanonical Ising dynamics: Problem: by making single spin flips it is basically impossible to keep the energy fixed for arbitrary K and J.

26. In this algorithm one probes the microstates of the system with energy E Microcanonical Ising dynamics: Creutz (1983) This is implemented by adding an auxiliary variable, called a demon such that system’s energy demon’s energy

27. Creutz algorithm: • Start with • Attempt to flip a spin: • accept the move if energy decreases • and give the excess energy to the demon. • if energy increases, take the needed energy from the • demon. Reject the move if the demon does not have • the needed energy.

28. Yields the caloric curve T(E). N=400, K=-0.35 E/N=-0.2416

29. To second order in ED the demon distribution is And it looks as if it is unstable for CV < 0 (particularly near the microcanonical tricritical point where CV vanishes). However the distribution is stable as long as the entropy increases with E (namely T>0) since the next to leading term is of order 1/N.

30. E M Breaking of Ergodicity in Microcanonical dynamics. Borgonovi, Celardo, Maianti, Pedersoli (2004); Mukamel, Ruffo, Schreiber (2005). Systems with short range interactions are defined on a convex region of their extensive parameter space. If there are two microstates with magnetizations M1 and M2 Then there are microstates corresponding to any magnetization M1 < M < M2 .

31. E M This is not correct for systems with long range interactions where the domain over which the model is defined need not be convex.

32. The available is not convex. Ising model with long and short range interactions m=M/N= (N+ - N-)/N u =U/N = number of broken bonds per site in a configuration corresponding to isolated down spins + + + - + + + + - + + - + + + + - + + Hence:

33. K=-0.4

34. m Local dynamics cannot make he system cross from one segment to another. Ergodicity is thus broken even for a finite system.

39. m Breaking of Ergodicity at finite systems has to do with the fact that the available range of m dcreases as the energy is lowered. Had it been then the model could have moved between the right and the left segments by lowering its energy with a probability which is exponentially small in the system size.

40. m1 ms 0 Time scales Relaxation time of a state at a local maximum of the entropy (metastable state) s

41. For a system with short range interactions is finite. It does not diverge with N. m=0 ms R m1 ms 0 critical radius above which droplet grows.

42. m1 ms 0 For systems with long range interactions the relaxation time grows exponentially with the system size. In this case there is no geometry. The dynamics depends only on m. The dynamics is that of a single particle moving in a potential V(m)=-s(m) Griffiths et al (1966) (CE Ising); Antoni et al (2004) (XY model); Chavanis et al (2003) (Gravitational systems)

43. M=0 is a local maximum of the entropy K=-0.4

44. 0 ms Relaxation of a state with a local minimum of the entropy (thermodynamically unstable) One would expect the relaxation time of the m=0 state to remain finite for large systems (as is the case of systems with short range interactions..

45. M=0 is a minimum of the entropy K=-0.25

46. One may understand this result by considering the following Langevin equation for m: With D~1/N

47. Fokker-Planck Equation: This is the dynamics of a particle moving in a double well potential V(m)=-s(m), with T~1/N starting at m=0.

48. Since D~1/N the width of the distribution is Taking for simplicity s(m)~am2, a>0, the problem becomes that of a Particle moving in a potential V(m) ~ -am2 at temperature T~D~1/N This equation yields at large t

49. Diverging time scales have been observed in a number of systems with log range interactions. The Hamiltonian Mean Field Model (HMF) (an XY model with mean field ferromagnetic interactions) There exists a class of quasistationary m=0 states with relaxation time Yamaguchi, Barre, Bouchet, Dauxois, Ruffo (2004)

50. Summary Some general thermodynamic and dynamical properties of system with long range interactions have been considered. Negative specific heat in microcanonical ensembles Canonical and microcanonical ensembles need not be equivalent whenever the canonical transition is first order. Breaking of ergodicity in microcanonical dynamics due to non-convexity of the domain over which the model exists. Long time scales, diverging with the system size. The results were derived for mean field long range interactions but they are expected to be valid for algebraically decaying potentials.