Thermodynamics and dynamics of systems with long range interactions David Mukamel

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

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

In order to make the energy and the entropy scale in the same way one may rescale the Hamiltonian (Kac prescription) This rescaling is relevant in a number of cases, such as self gravitating systems ferromagnetic dipolar systems

Self gravitating systems In finite systems, although E>>S, if T is high enough may be comparable to S, and the full free energy need to be considered.

Self gravitating systems e.g. in globular clusters: clusters of the order 10 5 stars within a distance of 10 2 light years. May be considered as a gas of massive objects Thus althoughsince T is large becomes comparable to S

Ferromagnetic dipolar systems v ~ 1/r 3 D is the shape dependent demagnetization factor

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

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 E 0 = xE 1 +(1-x)E 2 S 0 = xS 1 +(1-x)S 2 hence S is concave and the microcanonical specific heat is non-negative S

On the other hand in systems with long range interactions (non-additive), in the region E 1 <E<E 2 S E 0 = xE 1 +(1-x)E 2 S 0 xS 1 +(1-x)S 2 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

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 + _

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)

K/J0 T/J -1/2 1st order 2nd order K/J -1/ Ground state

Canonical (T,K) phase diagram

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

s=S/N, =E/N, m=M/N, u=U/N but Maximize to get

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.

m=0 discontinuity in T

Microcanonical phase diagram

canonicalmicrocanonical The two phase diagrams differ in the 1st order region of the canonical diagram

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

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

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

Creutz algorithm: 1.Start with 2.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.

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

To second order in E D the demon distribution is And it looks as if it is unstable for C V < 0 (particularly near the microcanonical tricritical point where C V 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.

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. E M If there are two microstates with magnetizations M 1 and M 2 Then there are microstates corresponding to any magnetization M 1 < M < M 2.

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

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: The available is not convex.

K=-0.4

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

Breaking of Ergodicity at finite systems has to do with the fact that the available range of m dcreases as the energy is lowered. m 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.

Time scales 0 m1m1 msms s Relaxation time of a state at a local maximum of the entropy (metastable state)

For a system with short range interactions is finite. It does not diverge with N. msms m=0 R 0 m1m1 msms critical radius above which droplet grows.

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) 0 m1m1 msms Griffiths et al (1966) (CE Ising); Antoni et al (2004) (XY model); Chavanis et al (2003) (Gravitational systems)

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

Relaxation of a state with a local minimum of the entropy (thermodynamically unstable) 0msms 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..

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

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

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.

This equation yields at large t Taking for simplicity s(m)~am 2, a>0, the problem becomes that of a particle moving in a potential V(m) ~ -am 2 at temperature T~D~1/N Since D~1/N the width of the distribution is

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)

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.

S. Ruffo J. Barre N. Schreiber A.Campa A. Giansanti K. Jain F. Bouchet

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.