Long-range systems with nonequivalent ensembles Hugo Touchette - PDF document

Long-range systems with nonequivalent ensembles Hugo Touchette National Institute for Theoretical Physics (NITheP) Stellenbosch, South Africa Long-range interacting many-body systems ICTP, Trieste, Italy 25-29 July 2016 UNIVERSITEIT STELLENBOSCH UNIVERSITY Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 1 / 20 Outline Thermodynamic F = E − TS – 1 Statistical ensembles 2 Thermodynamic equivalence 3 Macrostate equivalence Macrostates M ( ω ) 4 Microstate equivalence 5 Examples Microstates ω = ( ω 1 , . . . , ω N ) Referee B Ensemble inequivalence is not important, since systems with long-range forces do not evolve to equilibrium Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 2 / 20

Equilibrium statistical mechanics • N -particle system u • Microstate: ω = ( ω 1 , . . . , ω N ) • Hamiltonian: H ( ω ) • Macrostate: M ( ω ) • Ensemble: P u ( ω ) or P β ( ω ) • Closed or open system • Thermodynamic functions: s ( u ), f ( β ) T • Equilibrium states • Control parameters: u or β Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 3 / 20 Statistical ensembles Microcanonical ME Canonical CE • Parameter: u = H / N • Parameter: β = ( k B T ) − 1 • Microstate distribution: • Microstate distribution: � const H ( ω ) / N = u P β ( ω ) = e − β H ( ω ) P u ( ω ) = 0 otherwise Z ( β ) • Density of states: • Partition function: � � e − β H ( ω ) d ω Z ( β ) = Ω( u ) = δ ( H ( ω ) − uN ) d ω • Free energy: • Entropy: N →∞ − 1 1 ϕ ( β ) = lim N ln Z ( β ) s ( u ) = lim N ln Ω( u ) N →∞ • Equilibrium states: E u = { m u } • Equilibrium states: E β = { m β } Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 4 / 20

Equivalence of ensembles ME ? = CE Thermodynamic Macrostate Measure ? ? ? E u P u ← → E β ← → P β u ← → β ? s ( u ) ← → ϕ ( β ) • Short-range systems have equivalent ensembles • Long-range systems may have nonequivalent ensembles • All levels related to concavity of s ( u ) Short-range Long-range Small (finite) s s s u u u Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 5 / 20 Short- vs long-range interactions ε ε • Finite-range interaction • Interaction is ‘infinite’ range • Finite correlation length • Infinite correlation length • Extensive energy: U ∼ N • Non-extensive energy • Bulk dominates over surface • Bulk ∼ surface • Sub-system separation • No separation • Entropy always concave • Entropy possibly nonconcave Thermodynamics and statistical mechanics still defined Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 6 / 20

Concave entropy for short-range interactions • Entropy: 1 s ( u ) = lim N ln Ω N ( U = Nu ) N →∞ • Separation argument: s U , N U , N U , N 1 1 2 2 u 1 u 2 u U ≈ U 1 + U 2 s ( α u 1 + ¯ α u 2 ) ≥ α s ( u 1 ) + ¯ α s ( u 2 ) Ω N ( U 1 + U 2 ) ≥ Ω N 1 ( U 1 ) Ω N 2 ( U 2 ) Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 7 / 20 Two-block spin model [HT Am J Phys 2008] Referee A Entropy is always concave (at least I cannot imagine a counterexample) ↑ ↓ · · · ↑ ↑ ↑ . . . ↑ ↑ H N σ s 1 s 2 s N N � • Total energy: U = s i + N σ M1 M2 ln2 i =1 0.6 • Energy per spin: 0.4 s ( u ) u = U N ∈ [ − 2 , 2] 0.2 C G E • Entropy: 0 - 2 - 1 0 1 2 � s 0 ( u + 1) u ∈ [ − 2 , 0] u s ( u ) = s 0 ( u − 1) u ∈ (0 , 2] Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 8 / 20

Thermodynamic equivalence Microcanonical Canonical slope = u s ( u ) ϕ ( β ) slope = β u β s ( u ) = β u − ϕ ( β ) ϕ ( β ) = β u − s ( u ) ϕ ′ ( β ) = u s ′ ( u ) = β s ← → ϕ u ← → β s = ϕ ∗ ϕ = s ∗ Thermodynamic equivalence of ensembles Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 9 / 20 Thermodynamic nonequivalence s ϕ s ** u β u Non-concave Always concave Concave envelope s ϕ = s ∗ s ∗∗ = ϕ ∗ s � = ϕ ∗ = s ∗∗ • Thermodynamic nonequivalence of ensembles • Part of s ( u ) not recovered by ϕ ( β ) • Microcanonical properties not seen canonically Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 10 / 20

First-order phase transitions s ** ϕ u h u l u β β c u h u l u l u h β c β c u β β • s ( u ) nonconcave ⇒ ϕ ( β ) non-differentiable • First-order phase transition in canonical ensemble • Latent heat: ∆ u = u h − u l • Canonical skips over microcanonical Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 11 / 20 Macrostate equivalence Canonical Microcanonical Thermo u ↔ β • P u ( M N = m ) • P β ( M N = m ) M N ( ω ) Macro • E u = { m ∗ } Micro ( ω 1 , . . . , ω N ) • E β = { m ∗ } s ** s ** s s s u u u s = ϕ ∗ = s ∗∗ s � = ϕ ∗ = s ∗∗ Thermo level E u = E β E u � = E β Macrostate level Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 12 / 20

Mean-field Potts model [Costeniuc, Ellis & HT JMP 2005] • Hamiltonian: s ( u ) N H = − 1 � δ ω i ,ω j , ω i ∈ { 1 , 2 , 3 } 2 N i , j =1 • Distribution of spins: ν = ( a , b , b ) 1 1 1 − − − u 2 4 6 • Macrostate: a a = # spins 1 N -0.5 -0.4 -0.3 -0.2 • ME macrostate: a ( u ) u • CE macrostate: a ( β ) • Nonconcave entropy a • Nonequivalent ensembles • First-order canonical phase transition 0 2 4 6 8 10 β • Metastable states Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 13 / 20 Basic idea � const H ( ω ) / N = u P β ( ω ) = e − β H ( ω ) P u ( ω ) = Z ( β ) , 0 otherwise 1 Canonical with fixed energy = microcanonical P β ( ω | u ) = P u ( ω ) 2 Canonical = mixture of microcanonical � � P u ( m ) P β ( m ) = P β ( m | u ) P β ( u ) du = P β ( u ) du � �� � � �� � � �� � ME CE Bayes Theorem 3 Consequence: � E u E β = u ∈ U β � �� � Equilibrium energies 4 U β determined by concavity of s ( u ) Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 14 / 20

Measure equivalence • Microstate: ω = ( ω 1 , . . . , ω N ) Microcanonical Canonical � const H ( ω ) / N = u P β ( ω ) = e − β H ( ω ) P u ( ω ) = 0 otherwise Z ( β ) s ** s ** s s s u u u N ln P u ( ω ) N ln P u ( ω ) 1 1 lim P β ( ω ) = 0 lim P β ( ω ) � = 0 N →∞ N →∞ • P u ( ω ) ≈ P β ( ω ) • For almost all microstates Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 15 / 20 Recap Thermodynamic s ↔ ϕ u ↔ β E u = E β Macrostates s ′ ( u ) = β P u ( ω ) ≈ P β ( ω ) Microstates • Equivalence: s ( u ) concave • Nonequivalence: s ( u ) nonconcave • Valid for any macrostate • Energy constraint can be replaced by other constraints Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 16 / 20

Other ensembles Stretching [Cluzel et al Science 1996, Sinha & Samuel PRE 2005] x • Isotensional ensemble: • F = const F • x fluctuates x • Isometric ensemble: • x = const • F fluctuates F Graphs [Squartini et al PRL 2015] • Ensemble of graphs: P ( G ) • Fixed node number • Fixed degree sequence: { k 1 , k 2 , . . . } • Fixed distribution of degrees Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 17 / 20 Generalized ensembles [Costeniuc, Ellis, HT & Turkington JSP 2005] Generalized canonical ensemble Canonical ensemble � � e − β U − Ng ( U / N ) e − β U Z g ( β ) = Z ( β ) = ω ω N →∞ − 1 N →∞ − 1 ϕ ( β ) = lim N ln Z ( β ) ϕ g ( β ) = lim N ln Z g ( β ) s = ϕ g ∗ + g s � = ϕ ∗ • Recover equivalence with modified Legendre transform • Gaussian ensemble: g ( u ) = γ u 2 • Betrag ensemble: g ( u ) = γ | u − u 0 | • Universal ensembles: equivalence recovered with γ → ∞ Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 18 / 20

Conclusion Fixed constraint Average constraint Q ( ω ) = e − β H ( ω ) P ( ω | H = u ) Conditioning (micro) Exponential tilting (cano) • Asymptotic equivalence of distributions • Many Q equivalent to P More physical problems • What interactions lead to nonequivalent ensembles? • Can we experimentally measure nonconcave entropies? Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 19 / 20 References HT, R.S. Ellis, B. Turkington An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles Physica A 340, 138-146, 2004 General equivalence and nonequivalence of ensembles: Thermodynamic, macrostate, and measure levels J. Stat. Phys. 159, 987, 2015 Ensemble equivalence for general many-body systems Europhys. Lett. 96, 50010, 2011 Simple spin models with non-concave entropies Am. J. Phys. 76, 26, 2008 Methods for calculating nonconcave entropies J. Stat. Mech. P05008, 2010 Hugo Touchette (NITheP) Nonequivalent ensembles July 2016 20 / 20