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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

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Nonadditivity in the quasi-equilibrium state

  • f a short-range interacting system

Takashi Mori (Univ. Tokyo) Conference on Long-Range-Interacting Many Body Systems: from Atomic to Astrophysical Scales

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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

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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

𝐡 + 𝐼𝐢 + 𝐼 𝐡𝐢

𝐼

𝐡, 𝐼𝐢 ≫ 𝐼 𝐡𝐢

? The value of the Hamiltonian

depends on the microscopic state

3

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Interaction between subsystems

𝐼𝐡 eq, 𝐼𝐢 eq ≫ 𝐼𝐡𝐢 eq

There is a model in which this condition is satisfied but the two subsystems are not independent

𝐼𝐡, 𝐼𝐢 ≫ 𝐼𝐡𝐢 for any microscopic state

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Not sufficient There is a model in which this condition is violated but the two subsystems are almost independent Not necessary

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Definition of additivity in this talk

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A B A B

𝐼𝑗 = 𝐼𝐡 + 𝐼𝐢 + 𝐼𝐡𝐢 𝐼

𝑔 = 𝐼𝐡 + 𝐼𝐢

Quasi-static adiabatic process

𝐼 𝑒 = 𝐼𝐡 + 𝐼𝐢 + πœ‡πΌπ΅πΆ πœ‡: 1 β†’ 0 very slowly Amount of work performed by the system: 𝑋 = 𝐹 βˆ’ 𝐹′ If 𝑋 = 𝑝(π‘Š), the system is said to be additive

  • T. Mori, J. Stat. Phys. 159, 172 (2015)
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Consequence of additivity 1

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Additivity of entropy: 𝑇𝐡+𝐢 𝐹 = max

𝐹=𝐹𝐡+𝐹𝐢[𝑇𝐡(𝐹𝐡) + 𝑇𝐢(𝐹𝐢)] + 𝑝(π‘Š)

𝑋 = 𝑝(π‘Š)

  • T. Mori, J. Stat. Phys. 159, 172 (2015)

Entropy is conserved during a quasi-static adiabatic process 𝑇𝐡+𝐢 𝐹 = max

𝐹′=𝐹𝐡+𝐹𝐢

𝑇𝐡 𝐹𝐡 + 𝑇𝐢 𝐹𝐢 𝐹′ = 𝐹 βˆ’ 𝑋: the internal energy after the thermodynamic process 𝑋 = 𝑝 π‘Š β†’ 𝐹′ = 𝐹 + 𝑝(π‘Š)

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Consequence of additivity 2

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Additivity of entropy: 𝑇𝐡+𝐢 𝐹 = max

𝐹=𝐹𝐡+𝐹𝐢[𝑇𝐡(𝐹𝐡) + 𝑇𝐢(𝐹𝐢)] + 𝑝(π‘Š)

𝑋 = 𝑝(π‘Š)

Shape-independence of entropy

  • T. Mori, J. Stat. Phys. 159, 172 (2015)

𝑇𝐡+𝐢 𝑇𝐡+𝐢′ 𝑇𝐡 + 𝑇𝐢

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Consequence of additivity 3

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Additivity of entropy: 𝑇𝐡+𝐢 𝐹 = max

𝐹=𝐹𝐡+𝐹𝐢[𝑇𝐡(𝐹𝐡) + 𝑇𝐢(𝐹𝐢)] + 𝑝(π‘Š)

𝑋 = 𝑝(π‘Š)

Shape-independence of entropy Concavity of entropy

  • T. Mori, J. Stat. Phys. 159, 172 (2015)

π‘Š

𝐡

π‘Š = 𝑦, π‘Š

𝐢

π‘Š = 1 βˆ’ 𝑦 𝑑 𝜁 β‰₯ 𝑦𝑑 𝜁𝐡 + (1 βˆ’ 𝑦)𝑑(𝜁𝐢) 𝑑𝐡(𝜁) = 𝑑𝐢(𝜁) = 𝑑𝐡+𝐢(𝜁) = 𝑑(𝜁) for any 𝜁𝐡 and 𝜁𝐢 with 𝜁 = π‘¦πœπ΅ + 1 βˆ’ 𝑦 𝜁𝐢

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Consequence of additivity 4

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Additivity of entropy: 𝑇𝐡+𝐢 𝐹 = max

𝐹=𝐹𝐡+𝐹𝐢[𝑇𝐡(𝐹𝐡) + 𝑇𝐢(𝐹𝐢)] + 𝑝(π‘Š)

𝑋 = 𝑝(π‘Š)

Shape-independence of entropy Concavity of entropy

  • T. Mori, J. Stat. Phys. 159, 172 (2015)

𝑑𝐡(𝜁) = 𝑑𝐢(𝜁) = 𝑑𝐡+𝐢(𝜁) = 𝑑(𝜁) Ensemble equivalence, non-negativity of the specific heat,…

All the desired properties of additive systems are derived from the single condition!

𝑑 πœ‡πœπ΅ + 1 βˆ’ πœ‡ 𝜁𝐢 β‰₯ πœ‡π‘‘ 𝜁𝐡 + 1 βˆ’ πœ‡ 𝑑(𝜁𝐢)

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Nonadditive systems

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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

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Rigorous results in equilibrium statistical mechanics

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Any short-range interacting particle or spin systems with sufficiently strong short-range repulsions are additive Nonadditivity cannot be realized in an equilibrium state of a short-range interacting macroscopic system

No-Go theorem in equilibrium stat. mech.

π‘Š 𝑠 ≲ 1 𝑠𝑒+πœ— with some πœ— > 0 Short-range interaction

  • D. Ruelle, β€œstatistical mechanics”
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Possible ways towards long- range effective Hamiltonian

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Macroscopic systems Small systems

Non-neutral Coulomb system Dipolar systems Interaction range ~ system size

Non-equilibrium states

Nonequilibrium steady states (NESS): broken detailed balance condition Quasi-equilibrium states (metastable equilibrium): detailed balance satisfied

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Quasi-equilibrium state (metastable equilibrium)

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π‘Š(𝑠) ΰ·¨ π‘Š(𝑠)

π‘“βˆ’π›Ύΰ·©

π‘Š 𝑠

Quasi-equilibrium state is described by the equilibrium distribution

  • f an effective Hamiltonian
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Model

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Classical particle systems in the two or three dimensional space (In this talk: two-dimensional system) Each particle has an internal degree of freedom 𝜏 = Β±1 Pair interactions: π‘Š(𝑠)

𝑠 π‘Š(𝑠) 𝑆𝑗 + π‘†π‘˜ βˆ’π‘Š

Atomic radius 𝑆 Potential depth π‘Š Depending on 𝜏, the radius of the particle changes 𝑆 = 𝑆(𝜏), 𝑆 βˆ’1 < 𝑆(+1)

𝑆𝑗 π‘†π‘˜

𝑆𝑗 = 𝑆 πœπ‘— , π‘†π‘˜ = 𝑆(𝜏

π‘˜)

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Hamiltonian

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𝐼 = ෍

𝑗=1 𝑂 𝒒𝑗 2

2𝑛 + ෍

𝑗<π‘˜ 𝑂

π‘Š

πœπ‘—,πœπ‘˜(𝒓𝑗 βˆ’ π’“π‘˜) βˆ’ β„Ž ෍ 𝑗=1 𝑂

πœπ‘—

𝑠 𝑆(πœπ‘—) + 𝑆(𝜏

π‘˜)

βˆ’π‘Š π‘Š

πœπ‘—,πœπ‘˜(𝑠)

dynamics

{𝒓𝑗, 𝒒𝑗} : Hamilton dynamics πœπ‘— : Monte-Carlo dynamics

A model for Spin-Crossover material: P. GΓΌtlich, et.al., Angew.

Chem., Int. Ed. Engl. 33, 2024 (1994)

𝜏 = 1 High-Spin state 𝜏 = βˆ’1 Low-Spin state The size difference between HS and LS molecules is an experimental fact Canonical: Metropolis Microcanonical: Creutz

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Initial state

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triangular lattice structure put in the infinitely extended space π‘™πΆπ‘ˆ β‰ͺ π‘Š

0 This lattice structure is stable up to the time 𝜐~𝑓

π‘Š0 π‘™πΆπ‘ˆ

The system will reach the quasi-equilibrium state with this lattice structure held kept

𝑠 𝑏 βˆ’π‘Š π‘Š

πœπ‘—,πœπ‘˜(𝑠)

𝑏

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Intermediate state 1

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triangular lattice structure put in the infinitely extended space The system will reach the quasi-equilibrium state with this lattice structure held kept π‘™πΆπ‘ˆ β‰ͺ π‘Š

0 This lattice structure is stable up to the time 𝜐~𝑓

π‘Š0 π‘™πΆπ‘ˆ

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Intermediate state 2

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triangular lattice structure put in the infinitely extended space The system will reach the quasi-equilibrium state with this lattice structure held kept

𝑠 βˆ’π‘Š

π‘™πΆπ‘ˆ β‰ͺ π‘Š

0 This lattice structure is stable up to the time 𝜐~𝑓

π‘Š0 π‘™πΆπ‘ˆ

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Final state

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

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Numerical simulation

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𝑛 = 1, π‘™πΆπ‘ˆ = 0.26, β„Ž = 0, 𝑆 βˆ’1 = 1, 𝑆 1 = 1.1

two-step relaxation

𝐹/𝑂

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Quasi-equilibrium state

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Maxwell distribution

Momentum distribution in the quasi-equilibrium state

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Effective Hamiltonian

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In the quasi-equilibrium state, the lattice structure is maintained. οƒ We can approximate the interaction potential between the nearest neighbor pair by the quadratic one

𝑠 𝑆(πœπ‘—) + 𝑆(𝜏

π‘˜)

βˆ’π‘Š π‘Š

πœπ‘—,πœπ‘˜(𝑠)

ΰ·¨ π‘Š

πœπ‘—,πœπ‘˜(𝑠)

෩ 𝐼 = ෍

𝑗=1 𝑂 𝒒𝑗 2

2𝑛 + ෍

<𝑗,π‘˜> 𝑂

𝑙 2 𝒓𝑗 βˆ’ π’“π‘˜ βˆ’ 𝑆 πœπ‘— βˆ’ 𝑆 𝜏

π‘˜ 2

βˆ’ β„Ž ෍

𝑗=1 𝑂

πœπ‘—

  • nly the nearest neighbors

(only for nearest neighbors)

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Thermodynamic properties

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negative specific heat negative susceptibility

Red: average in the quasi-equilibrium state Green: average in the equilibrium state of ෩ 𝐼

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Thermodynamic properties

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negative specific heat negative susceptibility

Red: average in the quasi-equilibrium state Green: average in the equilibrium state of ෩ 𝐼

Nonadditivity in quasi-equilibrium states!

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Direct evidence of nonadditivity

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A B A B

Hamiltonian: ෩ 𝐼 Macroscopic amount

  • f work is necessary

𝑋 = 𝒫(𝑂)

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Thermal average of the interaction energy is negligible

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A B

෩ 𝐼 = ෍

𝑗=1 𝑂 𝒒𝑗 2

2𝑛 + ෍

<𝑗,π‘˜> 𝑂

𝑙 2 𝒓𝑗 βˆ’ π’“π‘˜ βˆ’ 𝑆 πœπ‘— βˆ’ 𝑆 𝜏

π‘˜ 2

βˆ’ β„Ž ෍

𝑗=1 𝑂

πœπ‘— Only nearest-neighbor interactions

𝐼

𝐡 eq, 𝐼𝐢 eq ≫ 𝐼 𝐡𝐢 eq

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Effective spin-spin interactions

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The degrees of freedom {𝒓𝑗, 𝒒𝑗, πœπ‘—} integrate out over {𝒓𝑗, 𝒒𝑗} Effective spin-spin interaction ΰ·© 𝐼 𝒓𝑗, 𝒒𝑗, πœπ‘— β†’ 𝐼eff πœπ‘— π‘“βˆ’π›ΎπΌeff ∼ ∫ π‘’π’“βˆ« π‘’π’’π‘“βˆ’π›Ύ ΰ·©

𝐼

It is difficult to obtain 𝐼eff οƒ  guess the interaction potential under the ansatz 𝐼eff = ෍

𝑗<π‘˜

πΎπ‘—π‘˜πœπ‘—πœ

π‘˜ βˆ’ β„Ž ෍ 𝑗=1 𝑂

πœπ‘—

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Effective spin-spin interactions

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The data of correlation functions: π·π‘—π‘˜ = βŒ©πœπ‘—πœ

π‘˜βŒͺ

π›ΎπΎπ‘—π‘˜ ≃ πœ€π‘—π‘˜ βˆ’ π·βˆ’1 π‘—π‘˜ πΎπ‘—π‘˜ = 1 𝑀𝑒 𝜚 𝒔𝑗 βˆ’ π’”π‘˜ 𝑀 It is found that πΎπ‘—π‘˜ obeys the scaling πΎπ‘—π‘˜ is independent of the temperature (energetic origin, not entropic origin) π‘€π‘’πΎπ‘—π‘˜ 𝑠

π‘—π‘˜/𝑀

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Meaning of the scaling

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πΎπ‘—π‘˜ = 1 𝑀𝑒 𝜚 𝒔𝑗 βˆ’ π’”π‘˜ 𝑀 The interaction range is comparable with the system size The interaction between two particles is very weak οƒ The interaction energy per particle is independent of the system size

Long-range interactions with Kac’s prescription extensive but nonadditive

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Discussion: Kac’s prescription

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The spin degrees of freedom of the model is described by the effective Hamiltonian with pair interactions πΎπ‘—π‘˜ = 1 𝑀𝑒 𝜚 𝒔𝑗 βˆ’ π’”π‘˜ 𝑀 Kac’s prescription naturally appears Originally short-range interacting systems

෩ 𝐼 = ෍

𝑗=1 𝑂 𝒒𝑗 2

2𝑛 + ෍

<𝑗,π‘˜> 𝑂

𝑙 2 𝒓𝑗 βˆ’ π’“π‘˜ βˆ’ 𝑆 πœπ‘— βˆ’ 𝑆 𝜏

π‘˜ 2

βˆ’ β„Ž ෍

𝑗=1 𝑂

πœπ‘—

Long-range force οƒ  nonadditivity Nearest neighbor interaction οƒ  extensivity

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β€œEquilibrium” depends on timescale

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Practically, we cannot distinguish equilibrium and quasi-equilibrium. Container is modeled by potential barrier If we consider the Hamiltonian of all the atoms of the gas and the container, this state is not the true equilibrium state

Container is eroded and broken

Quasi-equilibrium state

𝐼 = ෍

𝑗

𝒒𝑗

2

2𝑛 + 𝑉 𝒓𝑗 + ෍

π‘—β‰ π‘˜

π‘Š(𝒓𝑗 βˆ’ π’“π‘˜)

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β€œEquilibrium” depends on timescale

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Feynman in β€œStatistical Mechanics” If all the β€œfast” things have happened and all the β€œslow” things not, the system is said to be in thermal equilibrium.

In this sense, practically we cannot distinguish equilibrium and quasi-equilibrium. The concept of β€œequilibrium” depends on the timescale! In a certain (not infinitely long) timescale, short-range systems can exhibit nonadditivity.

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Dynamics

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Spin-spin effective interaction is long-ranged, But the interactions spread with finite speed. In short-time dynamics, the system behaves as a short-range interacting system. It is expected that the dynamical phenomena in this system differ from those in usual long-range interacting systems.

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Summary

  • Short-range interacting macroscopic systems can exhibit

nonadditivity in their quasi-equilibrium states

  • Such quasi-equilibrium states do not depend on the detail
  • f the dynamical rule
  • Effective Hamiltonian contains long-range interactions
  • Kac’s prescription is not necessary (the size-dependent

scaling naturally appears in the effective potential)

  • Distinction between quasi-equilibrium and equilibrium is

rather subtle.

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  • T. Mori, J. Stat. Phys. 159, 172 (2015)
  • T. Mori, Phys. Rev. Lett. 111, 020601 (2013)