We construct an exact, closed-form Carnot cycle whose state space is the simplex of probability distributions on a finite source alphabet, and whose working fluid is the optimal D-ary Huffman code attached to that source. The hot reservoir is identified with the uniform distribution, which maximises the average codeword length over all sources on a given alphabet; the cold reservoir is identified with a dominant-symbol quasi-uniform family whose average codeword length approaches its minimum possible value from above as the rare-symbol probability vanishes. Two adiabatic legs, defined by transcodings that preserve the dominant symbol’s unit-length codeword, close the cycle at a work cost that vanishes in the cold-reservoir limit. The resulting efficiency, expressed as the ratio of hot- to cold-reservoir temperatures in units of average codeword length, takes a closed form determined entirely by the alphabet sizes of the hot reservoir, and reproduces standard special cases including a value of eighty-seven and a half per cent for a binary source with two hundred and fifty-six symbols. We then embed the cycle in the Sagawa–Ueda generalised Jarzynski framework, identifying the trajectory-level mutual information with the log-likelihood difference between the source distribution and the codebook’s implicit distribution. Pathwise saturation of the resulting Jensen inequality is shown to require both source uniformity on its support and a matched codebook— a condition strictly stronger than the zero Kullback–Leibler criterion previously invoked in the literature. We derive the leading finite correction to the efficiency and show that it is linear in the rare-symbol weight, whereas the Jensen gap vanishes more slowly, so that pathwise saturation is restored before the Carnot limit is attained. The strict impossibility of reaching average codeword length equal to one for any source with more symbols than codeword letters is reinterpreted as a quantitative, third-law-style unattainability principle. The construction is exact at every step, is internally consistent with the Sagawa–Ueda framework, and closes rigorously the geometric Carnot diagram proposed in earlier work.
We study the extent of universality in the finite-size scaling function for the probability of minimal height in the stationary Abelian sandpile on lattice approximations of a region U ⊂ℂ with lattice spacing ε and with open boundary conditions. We show that under certain symmetry assumptions on the lattice, in the scaling limit ε→ 0 this probability at a point z ∈ U equals p_𝒢(0) + ε ^2 c_𝒢f_U(z) + O(ε ^3) , where p_𝒢(0) is the height 0 probability on the full lattice, c_𝒢> 0 is a constant depending only on the lattice, and f_U is a conformally covariant positive real-valued function depending only on U. This generalises a result of Brankov et al. [1], who considered the square lattice and the upper half plane. It also generalizes Example 4.10 in the recent study of the fermionic DGFF of Adame-Carillo and Ruszel [2]. We show via counterexamples that our symmetry assumption cannot be omitted, although conformal covariance may be recovered via an unusual scaling. In particular, the natural analogue of our result fails on general isoradial graphs. Our work is motivated by Jeng et al. [3], who computed the correction terms for all height variables in the special case of the upper half plane in ℤ^2 . They conjectured that their results hold more generally and the present paper is a step in an attempt to make this conjecture rigorous.
We propose an extended logistic equation with a generic fluctuating growth rate and solve the resulting random differential equation driven by bounded or unbounded non-white noise, using a previously reported functional method. This approach yields the exact evolution of the probability density function (PDF), moments and dispersion for different types of noise, namely, Gaussian colored noise (GCN), dichotomous noise, decaying noise, and Lévy noise, that may perturb the growth rate of a population. To address the problem of extrema densities for the logistic equation, we obtain exact analytical formulas for the maxima and minima densities, the first-passage time density, and the mean first-passage time for the particular case of Gaussian white noise (GWN). For the GCN, we further introduce an absorbing-boundary approximation that allows us to calculate analytically the maxima, minima, and first-passage time densities. Analytical predictions for the mean first-passage time are validated by Monte Carlo simulations in which the colored-noise trajectories are generated by an exact sampling scheme for the Ornstein-Uhlenbeck process, valid for any discretization step.
Mary Tsingou Menzel was interviewed on November 22, 2024, at her Los Alamos home, with the assistance of her daughter, Ann Menzel.
This paper considers the parameter estimation for a class of continuous-time stochastic dynamics driven by stochastic differential equations with multiple thresholds, based on discrete-time observations. The underlying state is characterized by state-dependent regime switching, where the drift function changes across different threshold regimes. To address the practically relevant case where both the number and locations of thresholds are unknown, we propose a sequential testing procedure to detect the presence of additional thresholds over a given interval, and develop a method to estimate the locations of thresholds and drift parameters. Building on these results, we propose a SNTD algorithm (sequential algorithm for nonlinear threshold dynamics) to jointly determine the number of thresholds, their locations, and the associated drift parameters. Theoretical guarantees for all proposed estimators are established, including convergence and asymptotic distributions. Simulation studies are provided to demonstrate the effectiveness of the proposed methods.
