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Employing a recent technology of tree surgery we prove a ``deletion-constriction'' formula for products of rooted spanning trees on weighted directed graphs that generalizes deletion-contraction on undirected graphs. The formula implies that, letting $\tau_x^\varnothing$, $\tau_x^+$, and $\tau_x^-$ be the rooted spanning tree polynomials obtained respectively by removing an edge in both directions or by forcing the tree to pass through either direction of that edge, the vectors $(\tau_x^\varnothing, \tau_x^+, \tau_x^-)$ are coplanar for all roots $x$. We deploy the result to give an alternative derivation of a recently found mutual linearity of stationary currents of Markov chains. We generalize deletion-constriction and current linearity among two edges, and conjecture that similar results may hold for arbitrary subsets of edges., Comment: 21 pages, 3 figures

For continuous-time Markov chains and open unimolecular chemical reaction networks, we prove that any two stationary currents are linearly related upon perturbations of a single edge's transition rates, arbitrarily far from equilibrium. We extend the result to non-stationary currents in the frequency domain, provide and discuss an explicit expression for the current-current susceptibility in terms of the network topology, and discuss possible generalizations. In practical scenarios, the mutual linearity relation has predictive power and can be used as a tool for inference or model proof-testing., Comment: Accepted for publication in Physical Review Letters

The nuclear two-photon or double-gamma ($2\gamma$) decay is a second-order electromagnetic process whereby a nucleus in an excited state emits two gamma rays simultaneously. To be able to directly measure the $2\gamma$ decay rate in the low-energy regime below the electron-positron pair-creation threshold, we combined the isochronous mode of a storage ring with Schottky resonant cavities. The newly developed technique can be applied to isomers with excitation energies down to $\sim100$\,keV and half-lives as short as $\sim10$\,ms. The half-life for the $2\gamma$ decay of the first-excited $0^+$ state in bare $^{72}\mathrm{Ge}$ ions was determined to be $23.9\left(6\right)$\,ms, which strongly deviates from expectations., Comment: Submitted to Physical Review Letters, 8 pages, 3 figures, 2 tables, 7 equations

Coarse-grained models are widely used to explain the effective behavior of partially observable physical systems with hidden degrees of freedom. Reduction procedures in state space typically disrupt Markovianity and a fluctuation relation cannot be formulated. A recently developed framework of transition-based coarse-graining gave rise to a fluctuation relation for a single current, while all others are hidden. Here, we extend the treatment to an arbitrary number of observable currents. Crucial for the derivation are the concepts of mixed currents and their conjugated effective affinities, that can be inferred from the time series of observable transitions. We also discuss the connection to generating functions, transient behavior, and how our result recovers the fluctuation relation for a complete set of currents., Comment: 37 pages, 11 figures

Lecture notes after the doctoral school (Post)Modern Thermodynamics held at the University of Luxembourg, December 2022, 5-7, covering and advancing continuous-time Markov chains, network theory, stochastic thermodynamics, large deviations, deterministic and stochastic chemical reaction networks, metastability, martingales, quantum thermodynamics, and foundational issues., Comment: Submission to SciPost

Stochastic chemical processes are described by the chemical master equation satisfying the law of mass-action. We first ask whether the dual master equation, which has the same steady state as the chemical master equation, but with inverted reaction currents, satisfies the law of mass-action, namely, still describes a chemical process. We prove that the answer depends on the topological property of the underlying chemical reaction network known as deficiency. The answer is yes only for deficiency-zero networks. It is no for all other networks, implying that their steady-state currents cannot be inverted by controlling the kinetic constants of the reactions. Hence, the network deficiency imposes a form of non-invertibility to the chemical dynamics. We then ask whether catalytic chemical networks are deficiency-zero. We prove that the answer is no when they are driven out of equilibrium due to the exchange of some species with the environment.

For continuous-time Markov chains we prove that, depending on the notion of effective affinity $F$, the probability of an edge current to ever become negative is either $1$ if $F< 0$ else $\sim \exp - F$. The result generalizes a ``noria'' formula to multicyclic networks. We give operational insights on the effective affinity and compare several estimators, arguing that stopping problems may be more accurate in assessing the nonequilibrium nature of a system according to a local observer. Finally we elaborate on the similarity with the Boltzmann formula. The results are based on a constructive first-transition approach., Comment: 25 pages, 2 figures

The infimum of an integrated current is its extreme value against the direction of its average flow. Using martingale theory, we show that the infima of integrated edge currents in time-homogeneous Markov jump processes are geometrically distributed, with a mean value determined by the effective affinity measured by a marginal observer that only sees the integrated edge current. In addition, we show that a marginal observer can estimate a finite fraction of the average entropy production rate in the underlying nonequilibrium process from the extreme value statistics in the integrated edge current. The estimated average rate of dissipation obtained in this way equals the above mentioned effective affinity times the average edge current. Moreover, we show that estimates of dissipation based on extreme value statistics can be significantly more accurate than those based on thermodynamic uncertainty ratios, as well as those based on a naive estimator obtained by neglecting nonMarkovian correlations in the Kullback-Leibler divergence of the trajectories of the integrated edge current., Comment: 38 pages, 6 figures

The modern thermodynamics of discrete systems is based on graph theory, which provides both algebraic methods to define observables and a geometric intuition of their meaning and role. However, because chemical reactions are usually many-to-many, chemical networks are rather described by hypergraphs, which lack a systematized algebraic treatment and a clear geometric intuition. Here we fill this gap by building fundamental bases of chemical cycles (encoding stationary behavior) and cocycles (encoding finite-time relaxation). We interpret them in terms of circulations and gradients on the hypergraph, and use them to properly identify nonequilibrium observables. As application, we unveil hidden symmetries in linear response and, within this regime, propose a reconstruction algorithm for large metabolic networks consistent with Kirchhoff's Voltage and Current Laws., Comment: 36 pages, 12 figures, 1 table; an appendix has been added

The fluctuation relation, a milestone of modern thermodynamics, is only established when a set of fundamental currents can be measured. Here we prove that it also holds for systems with hidden transitions if observations are carried ``at their own beat,'' that is, by stopping the experiment after a fixed number of visible transitions, rather than the elapse of an external clock time. This suggests that thermodynamic symmetries are more resistant to the loss of information when described in the space of transitions., Comment: 10 pages, 7 figures

Interpreting partial information collected from systems subject to noise is a key problem across scientific disciplines. Theoretical frameworks often focus on the dynamics of variables that result from coarse-graining the internal states of a physical system. However, most experimental apparatuses can only detect a partial set of transitions, while internal states are inaccessible. Here, we consider an observer who records a time series of occurrences of one or several transitions performed by a system, under the assumption that its underlying dynamics is Markovian. We pose the question of how one can use the transitions' information to make inferences of dynamical, thermodynamical, and biochemical properties. First, elaborating on first-passage time techniques, we derive analytical expressions for the probabilities of consecutive transitions and the time elapsed between them. Second, we derive a lower bound for the entropy production rate that equals the sum of two non-negative contributions, one due to the statistics of transitions and a second due to the statistics of inter-transition times. We also show that when only one current is measured, our estimate still detects irreversibility even in the absence of net currents. We illustrate the developed framework in experimentally-validated biophysical models of kinesin and dynein molecular motors, and in a minimal model for template-directed polymerization. Our results reveal that while entropy production is entailed in the statistics of two successive transitions of the same type, the statistics of two different successive transitions can probe the existence of an underlying disorder in the motion of a molecular motor. Taken all together, our results highlight the power of inference from transition statistics ranging from thermodynamic quantities to network-topology properties of Markov processes.

The structure of 106Pd, populated in the EC/beta+ decay of 106Ag, was investigated at the INFN Legnaro National Laboratories using the Spes Low-energy Internal Conversion Electron Spectrometer (SLICES). The K-internal conversion coefficients of some transitions have been measured and the electric monopole transition strengths, between low-lying 2+ and 0+ states, have been deduced. These experimental data combined with the results from internal conversion electron measurement on 104Pd previously performed by the Florence spectroscopy group were compared with the theoretical values calculated in the framework of the interacting proton-neutron boson model. Good agreement between experimental and theoretical values is found when interpreting the 0+3 level as an intruder state.

Several recent inequalities bound the precision of a current - counting net number of transitions in a system - by a thermodynamic measure of dissipation. However, while currents may be defined locally, dissipation is a global property. Inspired by the fact that ever since Carnot cycles are the unit elements of thermodynamic processes, we prove similar bounds tailored to cycle currents - counting net cycle completions - in terms of their conjugate affinities. We show that these inequalities are stricter than previous ones, even far from equilibrium, and that they allow to tighten those on transition currents. We illustrate our results with a simple model and discuss some technical and conceptual issues related to shifting attention from transition to cycle observables., Comment: 5 pages, 3 figures

We derive an expression for the minimal rate of entropy that sustains two reservoirs at different temperatures $T_0$ and $T_\ell$. The law displays an intuitive $\ell^{-1}$ dependency on the relative distance and a characterisic $\log^2 (T_\ell/T_0)$ dependency on the boundary temperatures. First we give a back-of-envelope argument based on the Fourier Law (FL) of conduction, showing that the least-dissipation profile is exponential. Then we revisit a model of a chain of oscillators, each coupled to a heat reservoir. In the limit of large damping we reobtain the exponential and squared-log behaviors, providing a self-consistent derivation of the FL. For small damping "equipartition frustration" leads to a well-known balistic behaviour, whose incompatibility with the FL posed a long-time challenge., Comment: Re-Submission to SciPost

Open systems may be perturbed out of equilibrium states either by subjecting them to nonconservative forces or by injecting external currents. For small perturbations, the linear response is quantified by two different matrices. In the framework of network thermodynamics, under very broad assumptions, we show that the two matrices are connected by a supersymmetry, predicting that they have the same spectrum - up to the degeneracy of the ground state. Our approach brings into the mathematics of supersymmetry a new ingredient, namely oblique projection operators., Comment: 5 pages, 2 figures

The response of thermodynamic systems perturbed out of an equilibrium steady-state is described by the reciprocal and the fluctuation-dissipation relations. The so-called fluctuation theorems extended the study of fluctuations far beyond equilibrium. All these results rely on the crucial assumption that the observer has complete information about the system. Such a precise control is difficult to attain, hence the following questions are compelling: Will an observer who has marginal information be able to perform an effective thermodynamic analysis? Given that such observer will only establish local equilibrium amidst the whirling of hidden degrees of freedom, by perturbing the stalling currents will he/she observe equilibrium-like fluctuations? We model the dynamics of open systems as Markov jump processes on finite networks. We establish that: 1) While marginal currents do not obey a full-fledged fluctuation relation, there exist effective affinities for which an integral fluctuation relation holds; 2) Under reasonable assumptions on the parametrization of the rates, effective and "real" affinities only differ by a constant; 3) At stalling, i.e. where the marginal currents vanish, a symmetrized fluctuation-dissipation relation holds while reciprocity does not; 4) There exists a notion of marginal time-reversal that plays a role akin to that played by time-reversal for complete systems, which restores the fluctuation relation and reciprocity. The above results hold for configuration-space currents, and for phenomenological currents provided that certain symmetries of the effective affinities are respected - a condition whose range of validity we deem the most interesting question left open to future inquiry. Our results are constructive and operational: we provide an explicit expression for the effective affinities and propose a procedure to measure them in laboratory., Comment: 41 pages. Comments are welcomed

We present a method for identification of models with good predictive performances in the family of Bayesian log-linear mixed models with Dirichlet process random effects. Such a problem arises in many different applications; here we consider it in the context of disclosure risk estimation, an increasingly relevant issue raised by the increasing demand for data collected under a pledge of confidentiality. Two different criteria are proposed and jointly used via a two-stage selection procedure, in a M-open view. The first stage is devoted to identifying a path of search; then, at the second, a small number of nonparametric models is evaluated through an application-specific score based Bayesian information criterion. We test our method on a variety of contingency tables based on microdata samples from the US Census Bureau and the Italian National Security Administration, treated here as populations, and carefully discuss its features. This leads us to a journey around different forms and sources of bias along which we show that (i) while based on the so called "score+search" paradigm, our method is by construction well protected from the selection-induced bias, and (ii) models with good performances are invariably characterized by an extraordinarily simple structure of fixed effects. The complexity of model selection - a very challenging and difficult task in a strictly parametric context with large and sparse tables - is therefore significantly defused by our approach. An attractive collateral result of our analysis are fruitful new ideas about modeling in small area estimation problems, where interest is in total counts over cells with a small number of observations., Comment: 32 pages, 7 figures

It is known that complementary oblique projections $\hat{P}_0 + \hat{P}_1 = I$ on a Hilbert space $\mathscr{H}$ have the same standard operator norm $\|\hat{P}_0\| = \|\hat{P}_1\|$ and the same singular values, but for the multiplicity of $0$ and $1$. We generalize these results to Hilbert spaces endowed with a positive-definite metric $G$ on top of the scalar product. Our main result is that the volume elements (pseudodeterminants $\det_+$) of the metrics $L_0,L_1$ induced by $G$ on the complementary oblique subspaces $\mathscr{H} = \mathscr{H}_0 \oplus \mathscr{H}_1$, and of those $\mathit{\Gamma}_0,\mathit{\Gamma}_1$ induced on their algebraic duals, obey the relations \begin{align} \frac{\det_+ L_1}{\det_+ \mathit{\Gamma}_0} = \frac{\det_+ L_0}{\det_+ \mathit{\Gamma}_1} = {\det}_+ G. \nonumber \end{align} Furthermore, we break this result down to eigenvalues, proving a "supersymmetry" of the two operators $\sqrt{\mathit{\Gamma}_0 L_0}$ and $\sqrt{L_1 \mathit{\Gamma}_1}$. We connect the former result to a well-known duality property of the weighted-spanning-tree polynomials in graph theory., Comment: 11 pages

Systems driven away from thermal equilibrium constantly deliver entropy to their environment. Determining this entropy production requires detailed information about the system's internal states and dynamics. However, in most practical scenarios, only a part of a complex experimental system is accessible to an external observer. In order to address this challenge, two notions of partial entropy production have been introduced in the literature as a way to assign an entropy production to an observed subsystem: one due to Shiraishi and Sagawa [Phys. Rev. E 91, 012130 (2015)] and another due to Polettini and Esposito [arXiv:1703.05715 (2017)]. We show that although both of these schemes provide a lower bound on the total entropy production, the latter -- which utilizes an effective thermodynamics description-- gives a better estimate of the total dissipation. Using this effective thermodynamic framework, we establish a partitioning of the total entropy production into two contributions that individually verify integral fluctuation theorems: an observable partial entropy production and a hidden entropy production assigned to the unobserved subsystem. Our results offer broad implications for both theoretical and empirical systems when only partial information is available.

Thermodynamics is usually formulated on the presumption that the observer has complete information about the system he/she deals with: no parasitic current, exact evaluation of the forces that drive the system. For example, the acclaimed Fluctuation Relation (FR), relating the probability of time-forward and time-reversed trajectories, assumes that the measurable transitions suffice to characterize the process as Markovian (in our case, a continuous-time jump process). However, most often the observer only measures a marginal current. We show that he/she will nonetheless produce an effective description that does not dispense with the fundamentals of thermodynamics, including the FR and the 2nd law. Our results stand on the mathematical construction of a hidden time reversal of the dynamics, and on the physical requirement that the observed current only accounts for a single transition in the configuration space of the system. We employ a simple abstract example to illustrate our results and to discuss the feasibility of generalizations., Comment: 8 pages, 1 figure

The widely debated feasibility of thermodynamic machines achieving Carnot efficiency at finite power has been convincingly dismissed. Yet, the common wisdom that efficiency can only be optimal in the limit of infinitely-slow processes overlooks the dual scenario of infinitely-fast processes. We corroborate that efficient engines at divergent power output are not theoretically impossible, framing our claims within the theory of Stochastic Thermodynamics. We inspect the case of an electronic quantum dot coupled to three particle reservoirs to illustrate the physical rationale., Comment: 5 pages, 2 figures

Small area models are mixed effects regression models that link the small areas and borrow strength from similar domains. When the auxiliary variables used in the models are measured with error, small area estimators that ignore the measurement error may be worse than direct estimators. Alternative small area estimators accounting for measurement error have been proposed in the literature but only for continuous auxiliary variables. Adopting a Bayesian approach, we extend the unit-level model in order to account for measurement error in both continuous and categorical covariates. For the discrete variables we model the misclassification probabilities and estimate them jointly with all the unknown model parameters. We test our model through a simulation study exploring different scenarios. The impact of the proposed model is emphasized through application to data from the Ethiopia Demographic and Health Survey where we focus on the women's malnutrition issue, a dramatic problem in developing countries and an important indicator of the socio-economic progress of a country., Comment: 25 pages, 3 figures