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Entropy as Noether charge for quasistatic gradient flow

  • Aaron Beyen ORCID logo EMAIL logo und Christian Maes
Veröffentlicht/Copyright: 7. Februar 2025
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Journal of Non-Equilibrium Thermodynamics
Aus der Zeitschrift Journal of Non-Equilibrium Thermodynamics Band 50 Heft 2

Abstract

Entropy increase is fundamentally related to the breaking of time-reversal symmetry. By adding the ‘extra dimension’ associated with thermodynamic forces, we extend that discrete symmetry to a continuous symmetry for the dynamical fluctuations around (nonlinear) gradient flow. The latter connects macroscopic equilibrium conditions upon introducing a quasistatic protocol of control parameters. The entropy state function becomes the Noether charge. As a result, and following ideas expressed by Shin-ichi Sasa and co-workers, the adiabatic invariance of the entropy, part of the Clausius heat theorem, gets connected with the Noether theorem.

Keywords: gradient flow; Noether’s theorem; large deviation theory; path space action
Corresponding author: Aaron Beyen, Department of Physics and Astronomy, Leuven, KU, Belgium, E-mail: 

Funding source: Research Foundation - Flanders (FWO)

Award Identifier / Grant number: 152725N

Acknowledgments

The authors thank Shin-ichi Sasa for private correspondence.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: Aaron Beyen is supported by the Research Foundation - Flanders (FWO) doctoral fellowship 1152725N.

  7. Data availability: Not applicable.

Appendix

A. Curie-Weiss dynamics II

As a final illustration, we continue with the kinetic Ising model on the complete graph, [54], [55], with energy function

(A1) E λ ( σ ) ≔ − N ϕ λ ( m N ( σ ) )

only depending on the magnetization (function), m N ( σ ) ≔ 1 N ∑ i = 1 N σ i , with ϕ λ (m) a smooth function whose exact form is irrelevant in what follows. We take the (finite) Curie-Weiss dynamics to be the Markov process on K N ≔{−1,+1} N with rates c λ (σ, k) for flipping σ = (σ 1, σ 2, …, σ k , …, σ N ) → σ k = (σ 1, σ 2, …, − σ k , …, σ N ), given by

(A2) c λ ( σ , k ) = ν ( β ) e β 2 ( E λ ( σ ) − E λ ( σ k ) )

where β represents the inverse temperature of the thermal bath,[5] and ν = ν(β) is a characteristic flip frequency possibly depending on β. These spin-flip rates satisfy detailed balance with respect to the potential (A1).

The backward generator L λ N of that Markov process is

L λ N f ( σ ) = ∑ k = 1 N a λ m N ( σ ) + b λ m N ( σ ) σ k f ( σ k ) − f ( σ ) a λ ( m N ( σ ) ) = ν ⁡ cosh β ϕ λ ′ ( m N ( σ ) ) + O 1 N b λ ( m N ( σ ) ) = − ν ⁡ sinh β ϕ λ ′ ( m N ( σ ) ) + O 1 N

To simplify the algebra, we use the (a λ , b λ ) parameterization as long as possible, but we remain in the Curie-Weiss model at all times. Note in particular that a λ (m) > b λ (m) for all m.

The Hamiltonian equals

(A3) H λ ( f ; m ) = sinh ( 2 f ) a λ ( m ) + m b λ ( m ) tanh ( f ) − b λ ( m ) + m a λ ( m )

See e.g., p109 in [50]. For the Lagrangian L ( m ̇ ; m ) , we put m ̇ = j , and we maximize f m ̇ − H λ ( f ; m ) with respect to f,

(A4) 0 = m ̇ − ∂ H λ ∂ f ( f ; m ) ⇒ f λ ( m ̇ ; m ) = 1 2 log ± m ̇ 2 + 4 1 − m 2 a λ 2 − b λ 2 + m ̇ 2 ( 1 − m ) ( a λ − b λ )

where we need the + sign to get a maximum since then

∂ 2 H λ ∂ f 2 ( f λ ( m ̇ ; m ) , m ) = 2 m ̇ 2 + 4 1 − m 2 a λ 2 − b λ 2 > 0

We have used that −1 ≤ m ≤ 1 and a λ (m) > b λ (m); in fact a λ 2 − b λ 2 = ν 2 . The corresponding Lagrangian becomes

(A5) L λ ( m , m ̇ ) = 1 2 − m ̇ 2 + 4 1 − m 2 ν 2 + m ̇ log m ̇ 2 + 4 1 − m 2 ν 2 + m ̇ 2 ( 1 − m ) ( a λ − b λ ) + 2 ( a λ + b λ m )

Moreover, one readily checks that the functions ψ λ , ψ λ ⋆ from Section 4.3 are given by

ψ λ ( m ̇ ; m ) = 1 2 2 1 − m 2 ν − m ̇ 2 + 4 ( 1 − m 2 ) ν 2 + m ̇ log m ̇ + m ̇ 2 + 4 ( 1 − m 2 ) ν 2 2 ν 1 − m 2 ψ λ ⋆ ( f , m ) = ν 1 − m 2 cosh ( 2 f ) − 1

The macroscopic equation of motion is obtained from L λ ( m , m ̇ ) = 0 , demanding that the logarithmic term and the other terms cancel separately:

(A6) m ̇ = G λ ( m ) ,    for  G λ ( m ) = − 2 ( b λ ( m ) + a λ ( m ) m )

which agrees with [56], [57]. The equilibrium solutions follow from m ̇ λ * = 0

(A7) G λ m λ * = 0 ⟺ m λ * = − b λ m λ * a λ m λ * = tanh ( β ϕ λ ′ m λ * )

As in (40), for the free energy in (6), we have detailed balance

(A8) L λ ( m , m ̇ ) − L λ ( m , − m ̇ ) = m ̇ a r c t a n h ( m ) + 1 2 log a λ + b λ a λ − b λ = m ̇ ∂ m F λ = d F λ d t

and F λ solves the stationary Hamilton-Jacobi equation,

(A9) H λ ( m , ∂ m F λ ) = 0

As in Section 5, we introduce a protocol λ → λ(ɛt) for t ∈ [ t 1 , t 2 ] = [ τ 1 ε , τ 2 ε ] , ε > 0 , with ɛ ↓ 0 making the process quasistatic. To be specific, we take a slowly-varying energy function ϕ λ → ϕ λ(ɛt) or equivalently a λ → a λ(ɛt), and b λ → b λ(ɛt), e.g. due to a time-dependent magnetic field h → h (ϵt), giving rise to the slowly varying Hamiltonian

(A10) H λ ( ε t ) ( f ; m ) = sinh ( 2 f ) ( a λ ( ε t ) + b λ ( ε t ) m ) tanh ( f ) − ( b λ ( ε t ) + a λ ( ε t ) m )

with Hamilton equations of motion

m ̇ = ∂ H λ ( ε t ) ∂ f = 2 ⁡ sinh ( 2 f ) a λ ( ε t ) + m b λ ( ε t ) − 2 ⁡ cosh ( 2 f ) b λ ( ε t ) + m a λ ( ε t ) f ̇ = − ∂ H λ ( ε t ) ∂ m = sinh ( 2 f ) a λ ( ε t ) + m a λ ( ε t ) ′ + b λ ( ε t ) ′ − tanh ( f ) b λ ( ε t ) + a λ ( ε t ) ′ + m b λ ( ε t ) ′

We denote the solutions to these equations as m t ( ε ) , f t ( ε ) with initial conditions m t = t 1 ( ε ) = m λ 1 * and f t = t 1 ( ε ) = f λ 1 * = 0 . Perturbatively in ɛ,

m t ( ε ) = m λ ( ε t ) * + ε Δ m t + O ( ε 2 ) f t ( ε ) = f λ ( ε t ) * + ε Δ f t + O ( ε 2 )

Here, the leading order term O (ɛ 0) corresponds to the equilibrium state m λ * but with λ replaced by λ(ɛt), i.e., an instantaneous equilibrium state. Up to order O (ɛ 2) we find

(A11) Δ m t = λ ̇ ( τ ) ∂ λ m λ ( τ ) * ∂ m G λ ( τ ) ( m λ ( τ ) * ) , with ∂ λ m * = − ∂ λ ( b λ / a λ ) 1 + ∂ m ( b λ / a λ )

(A12) Δ f t = 0

with G λ from (A6) and we have used that f λ * = 0 and thus also ∂ λ f λ * = 0 . Note that Δm t remains bounded only when ∂ m G λ ( m λ ( τ ) * ) does not diverge, which means we are not allowed to pass the critical point.

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Received: 2024-06-27
Accepted: 2025-01-17
Published Online: 2025-02-07
Published in Print: 2025-04-28

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Modeling high-pressure viscosities of fatty acid esters and biodiesel fuels based on modified rough hard-sphere-chain model and deep learning method
  4. Study on heat and mass transfer mechanism of unsaturated porous media under CW laser irradiation: with and without carrier gas
  5. Efficient ecological function optimization for endoreversible Carnot heat pumps
  6. Entropy as Noether charge for quasistatic gradient flow
  7. Numerical simulation of binary convection within the Soret regime in a tilted cylinder
  8. Is there a need for an extended phase definition for systems far from equilibrium?
  9. Existence of the Chapman–Enskog solution and its relation with first-order dissipative fluid theories
  10. Performance comparison of water towers and combined pumped hydro and compressed gas system and proposing a novel hybrid system to energy storage with a case study of a 50 MW wind farm
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https://doi.org/10.1515/jnet-2024-0054
Schlagwörter für diesen Artikel
gradient flow; Noether’s theorem; large deviation theory; path space action

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Modeling high-pressure viscosities of fatty acid esters and biodiesel fuels based on modified rough hard-sphere-chain model and deep learning method
  4. Study on heat and mass transfer mechanism of unsaturated porous media under CW laser irradiation: with and without carrier gas
  5. Efficient ecological function optimization for endoreversible Carnot heat pumps
  6. Entropy as Noether charge for quasistatic gradient flow
  7. Numerical simulation of binary convection within the Soret regime in a tilted cylinder
  8. Is there a need for an extended phase definition for systems far from equilibrium?
  9. Existence of the Chapman–Enskog solution and its relation with first-order dissipative fluid theories
  10. Performance comparison of water towers and combined pumped hydro and compressed gas system and proposing a novel hybrid system to energy storage with a case study of a 50 MW wind farm
  11. Thermodynamic characterization of transient valve temperatures in diesel engines using probabilistic methods
  12. Energetic analysis of a non-isothermal linear energy converter operated in reverse mode (I-LEC): heat pump
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