thermodynamics.
37 of 1538 NOTE documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| NOTE-011 v2 | Quantum Thermodynamics | A 2015–16 review with no new results, whose one synthesis (§5.2, §7) is that the field used four inequivalent notions of work: ensemble ⟨W⟩ = ∫tr[ρḢ]dt (Eq. 2), two-point-measurement W = E_m^(τ) − E_n^(0) (Eq. 24), ε-deterministic single-shot A^ε ≈ −kT ln(Z_Λ*/Z) (Eq. 49) and resource-theory w^max_ε = F^min_ε(ρ_S) − F(τ_S) (Eq. 50), of which only the last two coincide. It also judges that the field had “only made a small number of experimentally checkable predictions of new thermodynamic effects yet” (§7, p. 39). | Read |
| NOTE-032 v2 | Work capacity of channels with memory | The paper models agent and environment as coupled finite hidden-Markov channels. It defines the environment’s work capacity C_work = max over agent models of ⟨H(A_t|M_t) − H(S_t|M_t)⟩_t (in k_BT ln 2 per round) and proves two things. First, for unifilar environments whose percepts ignore actions, the efficient agents are exactly the randomize-and-forget ∩ maximally-predictive ones, and C_work = log|A| − h(S). Second, one binary memoryless environment exists where predictive, max-entropy-action and efficient agents form three nonempty, pairwise-disjoint sets: every predictive agent extracts ≤ 0, uniform-random agents extract 1 − h(3/4) ≈ 0.189 bits, and the optimum is ½·log₂(3/4 + 1/√2) ≈ 0.272 bits, reached by a memoryless agent that plays action 0 with probability 1/√2. | Read |
| NOTE-043 v2 | Observational entropy meets maximum entropy | The paper defines S^τ_M(ρ) = S(τ) − D_M(ρ‖τ) = −Σ_x p_x log(p_x/V_x), with V_x = Tr(τM_x)e^{S(τ)} (Eqs. 10–12). It is bounded as S(τ) ≥ S^τ_M(ρ) ≥ S(ρ) when ρ obeys the constraint S(ρ;τ) ≤ S(τ) (Eqs. 17–18). It reduces to 24 listed entropies, and it removes traditional observational entropy’s infinite-dimensional pathologies whenever S(τ) < ∞ (§VIII.7). Its “second laws” are bounds D_M(ρ̄‖τ) ≤ ε on the time-averaged state, split as equilibration plus prior-fit (Eq. 27). The paper proves the first term small in two regimes (Eqs. 28–29) and the second only under the ETH (Eq. 31). It states that in its own numerical examples the smallness “was [not] proved here analytically” (p. 12). | Read |
| NOTE-103 v2 | SEP — Philosophy of statistical mechanics | The entry argues that statistical mechanics has no canonical formalism but three umbrellas — Boltzmannian SM, the Boltzmann Equation and Gibbsian SM — and that every live account of the approach to equilibrium (ergodic, typicality, Mentaculus, long-run residence time, coarse-graining, interventionist, epistemic) survives Loschmidt’s and Zermelo’s objections only by giving up strict irreversibility or adding something outside the isolated dynamics; it closes on the unresolved “schism” that physicists compute in GSM while philosophers explain in BSM. | Read |
| NOTE-295 | Thermodynamics of Prediction | For a system driven by a stochastic signal through a fixed Markov kernel, the average work dissipated as the signal steps x_t → x_{t+1} equals, in units of k_BT, the system’s instantaneous nonpredictive information, I[s_t;x_t] − I[s_t;x_{t+1}] (Eq. 14). Summed over a protocol, this total “nostalgia” lower-bounds dissipation and excess work (Eq. 18), and it adds to Landauer’s bound: −β⟨Q⟩ ≥ I_e + I_mem − I_pred (Eq. 21). All three are averages over paths and protocols. None of them is a per-operation or per-bit floor on the cost of storing or using predictive information. | Read |
| NOTE-296 | Stochastic Thermodynamics of Learning | For a perceptron whose weights follow overdamped Langevin dynamics and whose predicted labels are fast two-state processes, the information learnt about fixed random labels is bounded by the weights’ total entropy production: Σ_μ I(σ_T^μ:σ^μ) ≤ Σ_n [ΔS(ω_n) + ΔQ_n] (Eq. 16, k_B = T = 1). This defines a learning efficiency η ≤ 1. The cost side includes the change in the weights’ own Shannon entropy, and in the paper’s toy model η → 0.99 as the heat ΔQ → 0 (slow driving). So the bound is not a heat floor of k_BT ln 2 per bit learnt. | Read |
| NOTE-310 | Notes on Landauer’s principle, reversible computation, and Maxwell’s Demon | Bennett states Landauer’s principle as a consequence of the Second Law. A logically irreversible step, meaning erasure or a merge of control flow, must export at least the lost entropy (k ln 2 per bit) to non-information-bearing degrees of freedom. That export need not take the form of heat (p. 1). Logically reversible operations, including copying onto a blank register and measurement, have no minimum cost. He calls “an intrinsic cost of order kT for every elementary act of information processing” a 20th-century “misconception” (pp. 5–6). Erasing random data can itself be thermodynamically reversible; the waste comes from applying a many-to-one operation to known data (pp. 1, 4). | Read |
| NOTE-413 | Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences | A finite-time fluctuation theorem for stochastic dynamics. If the dynamics are Markovian and microscopically reversible (path over reversed path = e^{−βQ}) and the entropy production ω = ln ρ(x_{−τ}) − ln ρ(x_{+τ}) − βQ is odd under time reversal, then P_F(+ω)/P_R(−ω) = e^{ω}. For processes that start and end in equilibrium ω = β(W − ΔF), so the Jarzynski equality is one line. For time-symmetric steady states, P(+ω)/P(−ω) = e^{ω} for any whole number of cycles. At long times it gives a heat fluctuation theorem and Gaussians with variance twice the mean. | Read |
| NOTE-415 | Life as a manifestation of the second law of thermodynamics | Proposes a “restated second law” (systems pushed from equilibrium resist the applied gradients by every available means) and reads Bénard cells, vortices, life and ecosystem succession as successive gradient destroyers, with the second law “necessary but not sufficient” for life. The law is asserted, not derived. The physics examples show organized structures raising dissipation, not that they must. The ecological evidence is one unreplicated marsh comparison and thermal-scanner and satellite correlations, one of whose tables is internally inconsistent. | Read |
| NOTE-416 | Statistical Physics of Adaptation | A macrostate fluctuation relation for driven systems, rearranged as an internal-entropy term, a reversal term and Ψ − Φ = −ln⟨e^{−βΔQ}⟩. It is read as a tendency of driven matter towards states formed by reliable work absorption and dissipation, “adaptation” without replication. The relation is exact. The tendency needs the other two terms held fixed, which the paper arranges only in contrived one-particle landscapes; the many-body claim and the “adaptive resonance” simulation are argued and reported, not shown. | Read |
| NOTE-417 | Nonequilibrium Equality for Free Energy Differences | The nonequilibrium work relation. Switch a classical system that starts in canonical equilibrium from H_0 to H_1 at any finite rate. Then ⟨e^{−βW}⟩ = e^{−βΔF} exactly, whatever the path or the switching time (Eq. 2). It is proved for an isolated Hamiltonian system by Liouville’s theorem, for a reservoir under weak coupling, and for a Nosé–Hoover thermostat with no extra assumption. Thermodynamic integration and the Zwanzig formula are its limits, ⟨W⟩ ≥ ΔF follows by Jensen, and Gaussian work gives W_diss = βσ²/2. Its practical reach is work fluctuations of order k_BT. | Read |
| NOTE-418 | Pattern formation outside of equilibrium | Read for the record’s question only. Patterns away from equilibrium are classified by the linear instability of a uniform state (type I_s, III_o, I_o) and described near onset by universal amplitude equations. There is in general no free energy or Lyapunov functional selecting them. Bénard’s own 1900 hexagons were surface-tension-driven, not Rayleigh’s buoyancy convection. Closed chemical systems must reach equilibrium, so their patterns are transients or need pumping. No global minimization principle exists beyond a perturbative one near threshold. | Skimmed |
| NOTE-419 | Statistical physics of self-replication | A coarse-grained second law, β⟨ΔQ⟩ + ln π(II→I) + ΔS_int ≥ 0, applied to self-replication: the heat a replicator releases per copy is bounded below by the log-odds against its copy falling apart within one generation, so durable, fast copies cost more heat. The general bound is sound. The E. coli and RNA numbers rest on an upper bound on the reverse probability that is argued from biology, not derived, and one printed logarithm has its argument inverted. | Read |
| NOTE-420 | Stochastic thermodynamics, fluctuation theorems and molecular machines | The 2012 reference review of stochastic thermodynamics. For Markovian dynamics coupled to baths of fixed temperature, work, heat and a stochastic entropy s = −ln p(x(τ),τ) are defined on single trajectories. Every known integral, detailed and Crooks-type fluctuation theorem follows from one master theorem, as the log-ratio of a path’s weight under the original and a conjugate dynamics. Dissipation measures how distinguishable forward and reverse paths are. Feedback adds the mutual information I to the second law (Sagawa–Ueda). Machine efficiency and efficiency at maximum power follow from a cycle decomposition. The integral theorems are not proofs of the second law, because the dynamics build irreversibility in. | Read |
| NOTE-422 | Comments on a derivation and application of the ‘maximum entropy production’ principle | Two corrections to Dewar. First, Dewar 2005 derives MaxEP far from equilibrium from a fluctuation theorem plus a quadratic approximation to the flux distribution, used as if valid for all f. Both can hold together only for small F, where A(F) = ∂λ/∂F is constant: the linear regime. So the orthogonality condition ∂D/∂F = 4λ, which yields both maximum and minimum dissipation, fails for nonlinear constitutive relations, as an exactly solvable spin chain shows. Second, Dewar 2003’s SOC divergence disappears once the quartic term is kept. | Read |
| NOTE-423 | Nonequilibrium thermodynamics and maximum entropy production in the Earth system | A review by MEP’s leading Earth-system advocate. MEP: given enough degrees of freedom, a system adopts the steady state of maximum entropy production. It is distinguished from Prigogine’s minimum principle, which describes relaxation toward a steady state and does not select among several. Two-box models show MEP arising from a flux–force trade-off. The Earth’s entropy budget is estimated and dominated by absorption of sunlight. MEP fits are surveyed from Paltridge’s climate model to sediment ecology, and life is argued to push the planet toward MEP. The author concedes MEP is not established. | Read |
| NOTE-424 v2 | Time, Structure and Fluctuations | Prigogine’s Nobel statement. Near equilibrium δ²S < 0 with ½∂δ²S/∂t = P > 0 makes δ²S a Lyapounov function, and in the strictly linear regime (constant L_ρρ′) the steady state minimizes entropy production. Far from equilibrium ½∂δ²S/∂t = ΣδJ_ρδX_ρ has no fixed sign, and ΣδJδX ≥ 0 is offered only as a sufficient stability condition. The Brusselator goes unstable at B > 1 + A². Dissipative structures are instabilities of the thermodynamic branch, triggered by fluctuations and sustained by exchange with the outside. The microscopic §§6–7 is a programme, linked to macroscopic thermodynamics only “in the linear region”. | Read |
| NOTE-425 | Energy flow and the organization of life | An essay arguing that life, beginning with the reductive citric-acid cycle, was forced into existence as a channel relieving geochemical free-energy stresses, like lightning or a hurricane. The framework is Jaynesian entropy maximization over states that include currents, not entropy-production principles. The emergence of life is a phase-transition “collapse” of a metastable abiotic earth. The structure of core metabolism is the only evidence. The authors state that the thesis is not quantitatively demonstrated. | Read |
| NOTE-426 | Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states | Jaynes’s maximum-entropy inference, applied to microscopic paths of an open stationary system, gives p_Γ = exp(A_Γ)/Z. The action is a symmetric end-point term plus τσ_Γ/2k_B. A fluctuation theorem p(σ) = e^{τσ/k_B} p(−σ) and ⟨σ⟩ ≥ 0 follow exactly. Maximum entropy production follows only in a mean-field approximation, and only if the number of paths increases with their irreversible action, which is assumed. The self-organized-criticality corollary is a mean-field divergence. As Grinstein and Linsker showed, it does not survive keeping the quartic term. | Read |
| NOTE-444 | An Organizational Account of Biological Functions | Functions are contributions to a differentiated self-maintaining organization by traits that organization produces and maintains (C1–C3). Closure alone grounds teleology and normativity but is shared by flames and hurricanes; differentiation is what makes functions meaningful. The account unifies “why is T there?” and “what is T for?” in current organization, and distinguishes functional, useful, dysfunctional, primary and secondary contributions. Reproduction is left to future “second-order” self-maintaining systems. | Read |
| NOTE-468 | Biological organisation as closure of constraints | A constraint acts on a process while being conserved at that process’s time scale; closure is a set of constraints, each directly dependent on and generative for another, that cannot be split. This separates closure from cycles of processes, grounds functions, and gives a boundary criterion, refined by a preliminary count-based tendency to closure for symbionts and by nested separated closures for levels. Applied only to schematic examples; closure is not claimed to define organisms. | Read |
| NOTE-708 | Vekilov — Nucleation | For lysozyme, measured homogeneous nucleation rates of crystals from solution are about ten orders of magnitude below the classical prediction, level off as supersaturation rises, and pass through a maximum as temperature falls. The paper explains these by two things classical theory lacks. A solution–crystal spinodal, defined kinetically as where the nucleus shrinks to one molecule, so the barrier vanishes. And a two-step mechanism, in which crystals nucleate inside metastable clusters of dense liquid that form fast; ordering inside the clusters is the slow step. For metastable clusters the two-step mechanism is inferred from a kinetic model fitted to one protein’s data, not observed. The evidence for other systems is weaker and the author grades it so. The paper also argues that near the spinodal the barrier is negligible, so heterogeneous substrates must act on the prefactor, not the barrier. | Read |
| NOTE-709 | Ostwald — Studies on the formation and transformation of solids, I | Ostwald times how small a crystal fragment can be and still seed a supercooled melt or supersaturated solution (down to about 10⁻⁹–10⁻¹² g). He shows that only the solid substance itself, or a strictly isomorphous solid, seeds. He divides states beyond equilibrium into metastable ones, which change only on contact with the other phase, and labile ones, where the new phase forms spontaneously. In §§32–41 he states the rule later called the rule of stages: on leaving a state, the form reached is “not the most stable … but the nearest” (pp. 306–307), meaning the form with the next-highest free energy. He offers it “not as a law” but as “a frequently applicable rule” (p. 308), drawn from experience and backed by an analogy with the van der Waals isotherm. The rule covers spontaneous formation only. Whether a metastable intermediate goes on to the next form depends on whether it is labile with respect to it; if it is merely metastable it stays until seeded (p. 308). The word “Stufenregel” does not appear in the paper. | Read |
| NOTE-711 | The Limiting Dynamics of SGD | SGD with momentum β, learning rate η, batch size S and weight decay λ is modelled as (η/2)(1 + β)θ̈ + (1 − β)θ̇ + λθ = −g(θ) + √(η/S) √Σ ξ(t). For linear regression with Σ = σ²H this is an OU process in phase space. Its stationary state is Gibbs on a modified loss, with inverse temperature κ = S(1 − β²), in broken detailed balance. Under a quadratic loss at the end of training the same holds for deep networks, and a ResNet-18 on ImageNet shows the predicted isotropy, velocity regularization, oscillations and speed. | Read |
| NOTE-714 | Noise Balance and Stationary Distribution of Stochastic Gradient Descent | SDE: dθ = −∇L dt + √(T C(θ)) dW, with T = η/S. With ℓ(u, w) = ℓ(λu, w/λ), d(‖u‖² − ‖w‖²)/dt = −T(uᵀC₁u − wᵀC₂w) (Theorem 1). The stationary point is Tr C(w) = Tr C(u). With weight decay γ the rate is TC₀ + γ (Corollary 2.1). Diagonal linear network ℓ = (Σᵢ ∏ₖ u_i^{(k)} x − y)²: sign coherence and balance for depth D > 1 (Theorem 3), exact stationary distribution (Theorem 4), collapse above T_c = β₂/α₃ for D = 1, tail ∝ v^{−5+3/(D+1)} independent of data. | Read |
| NOTE-788 v2 | Das, Manikandan and Banerjee — Information flow with hydrodynamic interactions | Two colloids are symmetrically coupled by the fluid, and the trap of one is moved by recorded Ornstein–Uhlenbeck noise. Mutual information, its time-delayed form and transfer entropy recover the order drive → driven particle → other particle. Under moderate drive they also show some backflow, which disappears under strong drive (numerics only). Stronger coupling lowers the information the driven particle shares with the drive and raises what the particles share. Without the drive, the coupled particles are statistically independent. | Read |
| NOTE-1370 | Kolchinsky and Wolpert — Semantic information and autonomous agency | Defines a system’s semantic information about its environment as the least syntactic information (initial mutual information, or transfer entropy over [0, τ]) that an intervention by coarse-graining the environment can leave while keeping the system’s viability at τ unchanged, viability being the negative Shannon entropy of the system’s state. Viability function, system–environment split, τ and initial distribution are the analyst’s choices, which the paper concedes and proposes to fix by maximizing the measures. One five-location food-seeking toy; no theorem beyond 0 ≤ semantic ≤ syntactic. | Read |
| NOTE-1443 | Mossio & Bich — What makes biological organisation teleological? | Biological organisation is intrinsically teleological because it self-determines, and it self-determines only as self-constraint: closure, a set of mutually dependent constraints that channel a far-from-equilibrium flow at a second level of causation. A cycle of transformations, such as the water cycle, is not closure. Dissipative structures do not realise closure; at most, on a contested reading, they make “a single macroscopic constraint” that acts on its own boundary conditions, and so would be minimally teleological but not organised and without functions. The paper never names differentiation or time scales as a condition: it folds the “division of labour” into closure. | Read |
| NOTE-1463 | Liang, De Los Rios & Busiello — Thermodynamic space of reaction networks | For any open mass-action reaction network at a stationary state, each reaction’s affinity is bounded by the extremal cycle affinities, per use of that reaction, over the elementary flux modes containing it, and each product of concentrations is bounded by the effective equilibrium constants of the extremal pathways that produce it. A fictitious, infinitely slow “chemical probe” reaction extends the bounds to species no reaction connects. The resulting thermodynamic space has zero width at equilibrium. Bistability (Schlögl), chiral symmetry breaking, dissipative self-assembly and pattern contrast all lie inside it, and measured concentrations outside it falsify a proposed topology. | Read |
| NOTE-1464 | Rao & Esposito — Thermodynamics of open chemical reaction networks | Builds the nonequilibrium thermodynamics of open, deterministic, mass-action networks of reversible elementary reactions on top of the rate equations, as stochastic thermodynamics is built on master equations. Chemostatting a species either breaks a conservation law or opens an emergent cycle; a network with no emergent affinity relaxes to a unique equilibrium, and a steady state dissipates the sum of emergent currents times emergent affinities. The minimal chemical work to hold a network away from equilibrium is a relative entropy of concentrations, a nonequilibrium Landauer principle. For complex-balanced networks the dissipation splits into a non-negative steady and a non-negative transient part. | Read |
| NOTE-1465 | Santolin et al. — Freidlin-Wentzell and stochastic thermodynamics | For overdamped diffusions with weak noise and a nonconservative force, the Freidlin–Wentzell quasi-potential is expanded about the detailed-balance free energy, each order integrated along the detailed-balanced relaxation, and the first order saturates the emergent second law. The weak-noise escape rate from an attractor is bracketed below by minus the entropy produced in relaxing into it and above by the entropy produced along the most likely escape path. The bracket closes at detailed balance and widens with the nonconservative force; in a sheared double well it stays usefully narrow for weak shear. | Read |
| NOTE-1468 | Remlein, Esposito & Avanzini — Open reaction networks emerging in closed ones | Shows that a closed reaction network behaves, to leading order, as an open chemostatted one when two conditions hold together: exchange reactions much faster than the rest (time-scale separation) and reservoir species much more abundant than the rest (abundance separation, a diverging chemical capacity). In that regime the chemical master equation, local detailed balance, entropy production and Gibbs balance of the open network all emerge. The regime lasts from the equilibration of the exchange reactions to a time of order the abundance ratio Ω, after which a driven network relaxes to the closed system’s equilibrium. Chemostats are emergent structures, not external idealisations. | Read |
| NOTE-1472 | Falasco & Esposito — macroscopic stochastic thermodynamics | Builds macroscopic stochastic thermodynamics from Markov jump processes with local detailed balance. Under extensive rates and potentials, a deterministic dynamics emerges whose Lyapunov function is the quasi-potential I_ss, and a large-deviation theory of fluctuations around it. Transition rates between attractors go as e^{−VΔI_ss} and are bracketed by the entropy produced along relaxation and escape paths; the bracket closes only near detailed balance. Gaussian-noise Langevin equations mis-estimate both the barriers and the entropy production. Read for those parts and the Schlögl example; field theory and two of three applications not read. | Skimmed |
| NOTE-1474 | Vellela & Qian — Schlögl model, stochastic and thermodynamic | The Schlögl model A + 2X ⇌ 3X, B ⇌ X, treated by rate equation, chemical master equation and nonequilibrium thermodynamics. Bistability needs the chemostats out of equilibrium; the master equation has one stationary distribution with two peaks, so in the long run which state the system occupies does not depend on where it started; the slow eigenvalue, exponentially small in volume, is the sum of the two escape rates; the more stable state can have the higher or the lower entropy production. The paper’s asymptotic rate formula carries a spurious factor 1/V, so the “breakdown” of asymptotics it reports at small volume is mostly that factor. | Read |
| NOTE-1476 | Dal Cengio, Lecomte & Polettini — Geometry of nonequilibrium reaction networks | Extends the Hill–Schnakenberg cycle analysis from linear networks (graphs) to interacting ones (hypergraphs) by row reduction of the stoichiometric matrix: cycles span its kernel and carry steady currents and non-conservative affinities, cocycles span the orthogonal complement and carry transient currents and conservative forces, with an integrator that recovers chemical potentials. One chemostat cannot drive a network; two can. Suppressing the reactions of a cocycle creates a new conserved quantity. Near equilibrium the relaxation and driven-response Onsager matrices share their spectrum, and the cycle decomposition gives a linear-regime reconstruction of metabolic fluxes that respects Kirchhoff’s laws. | Read |
| NOTE-1480 | Barato & Seifert — thermodynamic uncertainty relation | For counting currents in nonequilibrium steady states of finite Markov jump networks, the total entropy produced over time t times the squared relative uncertainty of the current is at least 2k_BT, whatever t. The letter proves this within linear response and, for unicyclic networks, at any affinity, with the sharper unicyclic bound (A/N) coth(A/2N); for multicyclic networks far from equilibrium it gives numerical evidence on seven networks and states the general case as a conjecture. | Read |
| NOTE-1481 | LaMont & Wiggins — thermodynamics and inference | Takes the evidence as a partition function with sample size N as inverse temperature, and reads off the thermodynamic derivatives. The internal energy is the leave-one-out predictive loss; the heat capacity becomes a “learning capacity” equal to K/2 in a regular model (equipartition) and to Watanabe’s learning coefficient in a singular one, and smaller when prior or discreteness fixes parameters (“freeze-out”). A prior making the Gibbs entropy zero reproduces AIC. The correspondence is formal; nothing in it is a physical current or dissipation. | Read |