physics

Nonequilibrium partition identity

Nonequilibrium partition identity is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Nonequilibrium partition identity rather than just read about it. In short: The nonequilibrium partition identity (NPI) is a remarkably simple and elegant consequence of the fluctuation theorem previously known as the Kawasaki identity: ⟨ exp ⁡ [ − Σ ¯ t t ] ⟩ = 1 , ∀ t {\displaystyle \left\langle {\exp[-{\overline {\Sigma }}_{t}\;t]}\right\rangle =1,\quad \forall t} (Carberry et al. 2004). Thus in spite of the second law inequality which might lead one to expect that the average would deca…

Key takeaways

Nonequilibrium partition identity belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Nonequilibrium partition identity to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Nonequilibrium partition identity from memory before moving on to harder problems.

Reference excerpt

The nonequilibrium partition identity (NPI) is a remarkably simple and elegant consequence of the fluctuation theorem previously known as the Kawasaki identity:

⟨ exp ⁡ [ − Σ ¯ t t ] ⟩ = 1 , ∀ t {\displaystyle \left\langle {\exp[-{\overline {\Sigma }}_{t}\;t]}\right\rangle =1,\quad \forall t}

(Carberry et al. 2004). Thus in spite of the second law inequality which might lead one to expect that the average would decay exponentially with time, the exponential probability ratio given by the FT exactly cancels the negative exponential in the average above leading to an average which is unity for all time. The first derivation of the nonequilibrium partition identity for Hamiltonian systems was by Yamada and Kawasaki in 1967. For thermostatted deterministic systems the first derivation was by Morriss and Evans in 1985.

Bibliography Kawasaki, Kyozi; Gunton, James D. (1973-10-01). "Theory of Nonlinear Transport Processes: Nonlinear Shear Viscosity and Normal Stress Effects". Physical Review A. 8 (4). American Physical Society (APS): 2048–2064. Bibcode:1973PhRvA...8.2048K. doi:10.1103/physreva.8.2048. ISSN 0556-2791. Yamada, Tomoji; Kawasaki, Kyozi (1967). "Nonlinear Effects in the Shear Viscosity of Critical Mixtures". Progress of Theoretical Physics. 38 (5). Oxford University Press (OUP): 1031–1051. Bibcode:1967PThPh..38.1031Y. doi:10.1143/ptp.38.1031. ISSN 0033-068X. Morriss, G.P.; Evans, Denis J. (1985-02-20). "Isothermal response theory". Molecular Physics. 54 (3). Informa UK Limited: 629–636. Bibcode:1985MolPh..54..629M. doi:10.1080/00268978500100481. ISSN 0026-8976. Carberry, D. M.; Williams, S. R.; Wang, G. M.; Sevick, E. M.; Evans, Denis J. (2004). "The Kawasaki identity and the Fluctuation Theorem" (PDF). The Journal of Chemical Physics. 121 (17). AIP Publishing: 8179–82. Bibcode:2004JChPh.121.8179C. doi:10.1063/1.1802211. hdl:1885/15803. ISSN 0021-9606. PMID 15511135.

See also Fluctuation theorem – Provides an equality that quantifies fluctuations in time averaged entropy production in a wide variety of nonequilibrium systems Crooks fluctuation theorem – Provides a fluctuation theorem between two equilibrium states; implies the Jarzynski equality

External links Marconi, U; Puglisi, A; Rondoni, L; Vulpiani, A (2008). "Fluctuation–dissipation: Response theory in statistical physics". Physics Reports. 461 (4–6). Elsevier BV: 111–195. arXiv:0803.0719. Bibcode:2008PhR...461..111M. doi:10.1016/j.physrep.2008.02.002. ISSN 0370-1573. S2CID 118575899.

Worked examples

Example 1 — a first encounter with Nonequilibrium partition identity

Start with the simplest possible case. Write down what Nonequilibrium partition identity claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Nonequilibrium partition identity before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Nonequilibrium partition identity ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Nonequilibrium partition identity

In research: Nonequilibrium partition identity appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Nonequilibrium partition identity in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Nonequilibrium partition identity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Equations, Non-equilibrium thermodynamics, Statistical mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Nonequilibrium partition identity outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Nonequilibrium partition identity in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nonequilibrium partition identity means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Nonequilibrium partition identity out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Nonequilibrium partition identity in simple terms?

The nonequilibrium partition identity (NPI) is a remarkably simple and elegant consequence of the fluctuation theorem previously known as the Kawasaki identity: ⟨ exp ⁡ [ − Σ ¯ t t ] ⟩ = 1 , ∀ t {\displaystyle \left\langle {\exp[-{\overline {\Sigma }}_{t}\;t]}\right\rangle =1,\quad \forall t} (Carb…

Why does Nonequilibrium partition identity matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Nonequilibrium partition identity?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Nonequilibrium partition identity.