Nonadditivity in the quasi-equilibrium state of a short-range - PowerPoint PPT Presentation

Nonadditivity in the quasi-equilibrium state of a short-range interacting system Takashi Mori (Univ. Tokyo) Conference on Long-Range-Interacting Many Body Systems: from Atomic to Astrophysical Scales

Outline 1. Introduction additivity and nonadditivity No-go theorem in equilibrium state Quasi-equilibrium states 2. Model 3. Numerical result non-additivity in the quasi-equilibrium state 4. Discussion and Summary 2

additivity and nonadditivity Rough meaning: If the system can be regarded as a collection of independent subsystems, the system is said to be additive. Expression in terms of the energy 𝐼 = 𝐼 𝐵 + 𝐼 𝐶 + 𝐼 𝐵𝐶 ? The value of the Hamiltonian 𝐼 𝐵 , 𝐼 𝐶 ≫ 𝐼 𝐵𝐶 depends on the microscopic state 3

Interaction between subsystems There is a model in which this 𝐼 𝐵 eq , 𝐼 𝐶 eq ≫ 𝐼 𝐵𝐶 eq condition is satisfied but the two subsystems are not Not sufficient independent 𝐼 𝐵 , 𝐼 𝐶 ≫ 𝐼 𝐵𝐶 for any microscopic state There is a model in which this condition is violated but the two subsystems are almost Not necessary independent 4

Definition of additivity in this talk T. Mori, J. Stat. Phys. 159, 172 (2015) Quasi-static adiabatic process A B A B 𝐼 𝑗 = 𝐼 𝐵 + 𝐼 𝐶 + 𝐼 𝐵𝐶 𝐼 𝑔 = 𝐼 𝐵 + 𝐼 𝐶 𝐼 𝑢 = 𝐼 𝐵 + 𝐼 𝐶 + 𝜇𝐼 𝐵𝐶 𝜇: 1 → 0 very slowly Amount of work performed by the system: 𝑋 = 𝐹 − 𝐹′ If 𝑋 = 𝑝(𝑊) , the system is said to be additive 5

Consequence of additivity 1 T. Mori, J. Stat. Phys. 159, 172 (2015) 𝑋 = 𝑝(𝑊) Entropy is conserved during a quasi-static adiabatic process 𝑇 𝐵+𝐶 𝐹 = max 𝑇 𝐵 𝐹 𝐵 + 𝑇 𝐶 𝐹 𝐶 𝐹 ′ =𝐹 𝐵 +𝐹 𝐶 𝐹 ′ = 𝐹 − 𝑋 : the internal energy after the thermodynamic process 𝑋 = 𝑝 𝑊 → 𝐹 ′ = 𝐹 + 𝑝(𝑊) Additivity of entropy: 𝑇 𝐵+𝐶 𝐹 = 𝐹=𝐹 𝐵 +𝐹 𝐶 [𝑇 𝐵 (𝐹 𝐵 ) + 𝑇 𝐶 (𝐹 𝐶 )] + 𝑝(𝑊) max 6

Consequence of additivity 2 T. Mori, J. Stat. Phys. 159, 172 (2015) 𝑋 = 𝑝(𝑊) Additivity of entropy: 𝑇 𝐵+𝐶 𝐹 = 𝐹=𝐹 𝐵 +𝐹 𝐶 [𝑇 𝐵 (𝐹 𝐵 ) + 𝑇 𝐶 (𝐹 𝐶 )] + 𝑝(𝑊) max Shape-independence of entropy 𝑇 𝐵 + 𝑇 𝐶 𝑇 𝐵+𝐶 𝑇 𝐵+𝐶′ 7

Consequence of additivity 3 T. Mori, J. Stat. Phys. 159, 172 (2015) 𝑋 = 𝑝(𝑊) Additivity of entropy: 𝑇 𝐵+𝐶 𝐹 = 𝐹=𝐹 𝐵 +𝐹 𝐶 [𝑇 𝐵 (𝐹 𝐵 ) + 𝑇 𝐶 (𝐹 𝐶 )] + 𝑝(𝑊) max Shape-independence of entropy 𝑡 𝐵 (𝜁) = 𝑡 𝐶 (𝜁) = 𝑡 𝐵+𝐶 (𝜁) = 𝑡(𝜁) Concavity of entropy 𝑊 𝑊 = 𝑦, 𝑊 𝐵 𝐶 𝑊 = 1 − 𝑦 𝑡 𝜁 ≥ 𝑦𝑡 𝜁 𝐵 + (1 − 𝑦)𝑡(𝜁 𝐶 ) for any 𝜁 𝐵 and 𝜁 𝐶 with 𝜁 = 𝑦𝜁 𝐵 + 1 − 𝑦 𝜁 𝐶 8

Consequence of additivity 4 T. Mori, J. Stat. Phys. 159, 172 (2015) 𝑋 = 𝑝(𝑊) Additivity of entropy: 𝑇 𝐵+𝐶 𝐹 = 𝐹=𝐹 𝐵 +𝐹 𝐶 [𝑇 𝐵 (𝐹 𝐵 ) + 𝑇 𝐶 (𝐹 𝐶 )] + 𝑝(𝑊) max Shape-independence of entropy 𝑡 𝐵 (𝜁) = 𝑡 𝐶 (𝜁) = 𝑡 𝐵+𝐶 (𝜁) = 𝑡(𝜁) Concavity of entropy 𝑡 𝜇𝜁 𝐵 + 1 − 𝜇 𝜁 𝐶 ≥ 𝜇𝑡 𝜁 𝐵 + 1 − 𝜇 𝑡(𝜁 𝐶 ) Ensemble equivalence, non- negativity of the specific heat,… All the desired properties of additive systems are derived from the single condition! 9

Nonadditive systems Entropy may depend on the shape of the system Entropy may be non-concave Ensemble equivalence may be violated Specific heat may be negative in the microcanonical ensemble etc … Unscreened long-range interactions make the system nonadditive 10

Rigorous results in equilibrium statistical mechanics 1 𝑊 𝑠 ≲ Short-range interaction 𝑠 𝑒+𝜗 with some 𝜗 > 0 Any short-range interacting particle or spin systems with sufficiently strong short-range repulsions are additive D. Ruelle , “statistical mechanics” Nonadditivity cannot be realized in an equilibrium state of a short-range interacting macroscopic system No-Go theorem in equilibrium stat. mech. 11

Possible ways towards long- range effective Hamiltonian Small systems Interaction range ~ system size Macroscopic systems Non-neutral Coulomb system Dipolar systems Non-equilibrium states Nonequilibrium steady states (NESS): broken detailed balance condition Quasi-equilibrium states (metastable equilibrium): detailed balance satisfied 12

Quasi-equilibrium state (metastable equilibrium) ෨ 𝑊(𝑠) 𝑊(𝑠) 𝑓 −𝛾෩ 𝑊 𝑠 Quasi-equilibrium state is described by the equilibrium distribution of an effective Hamiltonian 13

Model Classical particle systems in the two or three dimensional space (In this talk: two-dimensional system) 𝑊(𝑠) 𝑆 𝑘 Pair interactions: 𝑊(𝑠) 𝑆 𝑗 + 𝑆 𝑘 𝑆 𝑗 𝑠 Atomic radius 𝑆 Potential depth 𝑊 0 −𝑊 0 Each particle has an internal degree of freedom 𝜏 = ±1 Depending on 𝜏 , the radius of the particle changes 𝑆 = 𝑆(𝜏) , 𝑆 −1 < 𝑆(+1) 𝑆 𝑗 = 𝑆 𝜏 𝑗 , 𝑆 𝑘 = 𝑆(𝜏 𝑘 ) 14

Hamiltonian 𝑊 𝜏 𝑗 ,𝜏 𝑘 (𝑠) 𝑆(𝜏 𝑗 ) + 𝑆(𝜏 𝑘 ) 𝑂 𝒒 𝑗 𝑂 𝑂 2 𝐼 = ෍ 2𝑛 + ෍ 𝑊 𝜏 𝑗 ,𝜏 𝑘 (𝒓 𝑗 − 𝒓 𝑘 ) − ℎ ෍ 𝜏 𝑗 𝑠 𝑗=1 𝑗<𝑘 𝑗=1 −𝑊 0 dynamics A model for Spin-Crossover {𝒓 𝑗 , 𝒒 𝑗 } : Hamilton dynamics material: P. Gütlich, et.al., Angew. Chem., Int. Ed. Engl. 33, 2024 (1994) 𝜏 𝑗 : Monte-Carlo dynamics 𝜏 = 1 High-Spin state 𝜏 = −1 Low-Spin state Canonical: Metropolis The size difference between HS and Microcanonical: Creutz LS molecules is an experimental fact 15

Initial state triangular lattice structure put in the infinitely extended space 𝑊 𝜏 𝑗 ,𝜏 𝑘 (𝑠) 𝑏 𝑠 −𝑊 0 𝑏 𝑊0 𝑙 𝐶 𝑈 ≪ 𝑊 0 This lattice structure is stable up to the time 𝜐~𝑓 𝑙 𝐶 𝑈 The system will reach the quasi-equilibrium state with this lattice structure held kept 16

Intermediate state 1 triangular lattice structure put in the infinitely extended space 𝑊0 𝑙 𝐶 𝑈 ≪ 𝑊 0 This lattice structure is stable up to the time 𝜐~𝑓 𝑙 𝐶 𝑈 The system will reach the quasi-equilibrium state with this lattice structure held kept 17

Intermediate state 2 triangular lattice structure put in the infinitely extended space 𝑠 −𝑊 0 𝑊0 𝑙 𝐶 𝑈 ≪ 𝑊 0 This lattice structure is stable up to the time 𝜐~𝑓 𝑙 𝐶 𝑈 The system will reach the quasi-equilibrium state with this lattice structure held kept 18

Final state triangular lattice structure put in the infinitely extended space The particles finally go somewhere far away 19

Numerical simulation 𝑛 = 1, 𝑙 𝐶 𝑈 = 0.26, ℎ = 0, 𝑆 −1 = 1, 𝑆 1 = 1.1 two-step relaxation 𝐹/𝑂 20

Quasi-equilibrium state Momentum distribution in the quasi-equilibrium state Maxwell distribution 21

Effective Hamiltonian In the quasi-equilibrium state, the lattice structure is maintained.  We can approximate the interaction potential between the nearest neighbor pair by the quadratic one (only for nearest neighbors) 𝑊 𝜏 𝑗 ,𝜏 𝑘 (𝑠) ෨ 𝑊 𝜏 𝑗 ,𝜏 𝑘 (𝑠) 𝑆(𝜏 𝑗 ) + 𝑆(𝜏 𝑘 ) 𝑠 only the nearest neighbors −𝑊 0 𝑂 𝒒 𝑗 𝑂 𝑂 2 𝑙 2 ෩ 𝐼 = ෍ 2𝑛 + ෍ 𝒓 𝑗 − 𝒓 𝑘 − 𝑆 𝜏 𝑗 − 𝑆 𝜏 − ℎ ෍ 𝜏 𝑗 𝑘 2 𝑗=1 <𝑗,𝑘> 𝑗=1 22

Thermodynamic properties Red: average in the quasi-equilibrium state Green: average in the equilibrium state of ෩ 𝐼 negative specific heat negative susceptibility 23

Thermodynamic properties Red: average in the quasi-equilibrium state Green: average in the equilibrium state of ෩ 𝐼 negative specific heat negative susceptibility Nonadditivity in quasi-equilibrium states! 24

Direct evidence of nonadditivity A B A B Hamiltonian: ෩ 𝐼 𝑋 = 𝒫(𝑂) Macroscopic amount of work is necessary 25

Thermal average of the interaction energy is negligible A B 𝑂 𝒒 𝑗 𝑂 𝑂 2 𝑙 2 ෩ 𝐼 = ෍ 2𝑛 + ෍ 𝒓 𝑗 − 𝒓 𝑘 − 𝑆 𝜏 𝑗 − 𝑆 𝜏 − ℎ ෍ 𝜏 𝑗 𝑘 2 𝑗=1 <𝑗,𝑘> 𝑗=1 Only nearest-neighbor interactions 𝐼 𝐵 eq , 𝐼 𝐶 eq ≫ 𝐼 𝐵𝐶 eq 26

Effective spin-spin interactions The degrees of freedom {𝒓 𝑗 , 𝒒 𝑗 , 𝜏 𝑗 } integrate out over {𝒓 𝑗 , 𝒒 𝑗 } Effective spin-spin interaction ෩ 𝑓 −𝛾𝐼 eff ∼ ∫ 𝑒𝒓∫ 𝑒𝒒𝑓 −𝛾 ෩ 𝐼 𝐼 𝒓 𝑗 , 𝒒 𝑗 , 𝜏 𝑗 → 𝐼 eff 𝜏 𝑗 It is difficult to obtain 𝐼 eff  guess the interaction potential under the ansatz 𝑂 𝐼 eff = ෍ 𝐾 𝑗𝑘 𝜏 𝑗 𝜏 𝑘 − ℎ ෍ 𝜏 𝑗 𝑗<𝑘 𝑗=1 27