Coarse-grained Entropy Research Articles

Coarse-grained entropy production with multiple reservoirs: Unraveling the role of time scales and detailed balance in biology-inspired systems

A general framework to describe a vast majority of biology-inspired systems is to model them as stochastic processes in which multiple couplings are in play at the same time. Molecular motors, chemical reaction networks, catalytic enzymes, and particles exchanging heat with different baths, constitute some interesting examples of such a modelization. Moreover, they usually operate out of equilibrium, being characterized by a net production of entropy, which entails a constrained efficiency. Hitherto, in order to investigate multiple processes simultaneously driving a system, all theoretical approaches deal with them independently, at a coarse-grained level, or employing a separation of time-scales. Here, we explicitly take in consideration the interplay among time-scales of different processes, and whether or not their own evolution eventually relaxes toward an equilibrium state in a given sub-space. We propose a general framework for multiple coupling, from which the well-known formulas for the entropy production can be derived, depending on the available information about each single process. Furthermore, when one of the processes does not equilibrate in its sub-space, even if much faster than all the others, it introduces a finite correction to the entropy production. We employ our framework in various simple and pedagogical examples, for which such a corrective term can be related to a typical scaling of physical quantities in play.

Quantum correlation entropy

We study quantum coarse-grained entropy and demonstrate that the gap in entropy between local and global coarse-grainings is a natural generalization of entanglement entropy to mixed states and multipartite systems. This "quantum correlation entropy" $S^{\rm QC}$ is additive over independent systems, is invariant under local unitary operations, measures total nonclassical correlations (vanishing on states with strictly classical correlation), and reduces to the entanglement entropy for bipartite pure states. It quantifies how well a quantum system can be understood via local measurements, and ties directly to non-equilibrium thermodynamics, including representing a lower bound on the quantum part of thermodynamic entropy production. We discuss two other measures of nonclassical correlation to which this entropy is equivalent, and argue that together they provide a unique thermodynamically distinguished measure.

Outer entropy equals Bartnik-Bray inner mass and the gravitational ant conjecture

Entropy and energy are found to be closely tied on our quest for quantum gravity. We point out an interesting connection between the recently proposed outer entropy, a coarse-grained entropy defined for a compact spacetime domain motivated by the holographic duality, and the Bartnik-Bray quasilocal mass long known in the mathematics community. In both scenarios, one seeks an optimal spacetime fill-in of a given closed, connected, spacelike, codimension-two boundary. We show that for an outer-minimizing mean-convex surface, the Bartnik-Bray inner mass matches exactly with the irreducible mass corresponding to the outer entropy. The equivalence implies that the area laws derived from the outer entropy are mathematically equivalent as the monotonicity property of the quasilocal mass. It also gives rise to new bounds between entropy and the gravitational energy, which naturally gives the gravitational counterpart to Wall's ant conjecture. We also observe that the equality can be achieved in a conformal flow of metrics, which is structurally similar to the Ceyhan-Faulkner proof of the ant conjecture. We compute the small sphere limit of the outer entropy and it is proportional to the bulk stress tensor as one would expect for a quasilocal mass. Lastly, we discuss some implications of taking quantum matter into consideration in the semiclassical setting.

Classical dynamical coarse-grained entropy and comparison with the quantum version.

We develop the framework of classical observational entropy, which is a mathematically rigorous and precise framework for nonequilibrium thermodynamics, explicitly defined in terms of a set of observables. Observational entropy can be seen as a generalization of Boltzmann entropy to systems with indeterminate initial conditions, and it describes the knowledge achievable about the system by a macroscopic observer with limited measurement capabilities; it becomes Gibbs entropy in the limit of perfectly fine-grained measurements. This quantity, while previously mentioned in the literature, has been investigated in detail only in the quantum case. We describe this framework reasonably pedagogically, then show that in this framework, certain choices of coarse-graining lead to an entropy that is well-defined out of equilibrium, additive on independent systems, and that grows toward thermodynamic entropy as the system reaches equilibrium, even for systems that are genuinely isolated. Choosing certain macroscopic regions, this dynamical thermodynamic entropy measures how close these regions are to thermal equilibrium. We also show that in the given formalism, the correspondence between classical entropy (defined on classical phase space) and quantum entropy (defined on Hilbert space) becomes surprisingly direct and transparent, while manifesting differences stemming from noncommutativity of coarse-grainings and from nonexistence of a direct classical analog of quantum energy eigenstates.

The Science of $${\Theta \Delta }^{\text{cs}}$$

There is a long tradition of thinking of thermodynamics, not as a theory of fundamental physics (or even a candidate theory of fundamental physics), but as a theory of how manipulations of a physical system may be used to obtain desired effects, such as mechanical work. On this view, the basic concepts of thermodynamics, heat and work, and with them, the concept of entropy, are relative to a class of envisaged manipulations. This article is a sketch and defense of a science of manipulations and their effects on physical systems. I call this science thermo-dynamics (with hyphen), or \({\Theta \Delta }^{\text{cs}}\), for short, to highlight that it may be different from the science of thermodynamics, as the reader conceives it. An upshot of the discussion is a clarification of the roles of the Gibbs and von Neumann entropies. Light is also shed on the use of coarse-grained entropies.

Grain boundary diffusion in CoCrFeMnNi high entropy alloy: Kinetic hints towards a phase decomposition

Grain boundary diffusion of the principal elements 57Co, 51Cr, 59Fe and 54Mn in a coarse-grained equiatomic CoCrFeMnNi high entropy alloy is measured in a wide temperature range of 643 to 1273 K in both C- and B-type kinetic regimes after Harrison’s classification. The results suggest that the product of the pertinent segregation factors, s, and the grain boundary width, δ, is about 0.5 nm for all elements at temperatures T > 800 K. Whereas one short-circuit contribution is observed at higher temperatures above 800 K, the penetration profiles in the C-type kinetic regime (643 – 703 K) reveal two distinct contributions that hint towards a phase decomposition at a fraction of high-angle grain boundaries at these temperatures and the existence of a structural multiplicity of high-angle grain boundaries. A correlative microscopy combining transmission Kikuchi diffraction and atom probe tomography manifests formation of neighboring Ni-Mn-rich and Cr-rich precipitates at high angle grain boundaries. Transmission electron microscopy revealed an increased dislocation density in the vicinity of such interfaces which is suggested to be a reason of the enhanced diffusion rates at low temperatures for such short circuits.

Entropy, mutual information, and systematic measures of structured spiking neural networks

The aim of this paper is to investigate various information-theoretic measures, including entropy, mutual information, and some systematic measures that are based on mutual information, for a class of structured spiking neuronal networks. In order to analyze and compute these information-theoretic measures for large networks, we coarse-grained the data by ignoring the order of spikes that fall into the same small time bin. The resultant coarse-grained entropy mainly captures the information contained in the rhythm produced by a local population of the network. We first show that these information theoretical measures are well-defined and computable by proving stochastic stability and the law of large numbers. Then we use three neuronal network examples, from simple to complex, to investigate these information-theoretic measures. Several analytical and computational results about properties of these information-theoretic measures are given.

Redefinition of the Boltzmann–Gibbs–Shannon entropy in systems with continuous states

The Boltzmann–Gibbs–Shannon (BGS) entropy Sd of a system with discrete states is inherently non-negative, and attains its minimum value of zero only when the system is known with certainty to occupy a particular state. In contrast, the BGS entropy S of a system with continuous states can be negative with no lower bound. This disparity is traced to the fact that the generalisation of Sd to obtain S is based on an implicit assumption which becomes conceptually inconsistent when the continuous probability density varies too rapidly with . This analysis suggests an alternative generalisation of Sd which results in a fully consistent and inherently non-negative entropy . The resulting expression for is observed to be algebraically equivalent to a conventional coarse-grained BGS entropy, but with the essential difference that the previously arbitrary cell sizes are now well defined and are no longer ambiguous. The non-equilibrium time dependence of is well known to be thermodynamically anomalous, whereas that of is shown to be consistent with the expected behaviour of the thermodynamic entropy and its irreversible production rate in both conservative and dissipative systems with mixing behaviour.

From black hole entropy to energy-minimizing states in QFT

Behind certain marginally trapped surfaces one can construct a geometry containing an extremal surface of equal, but not larger area. This construction underlies the Engelhardt-Wall proposal for explaining the Bekenstein-Hawking entropy as a coarse-grained entropy. The construction can be proven to exist classically but fails if the null energy condition is violated. Here we extend the coarse-graining construction to semiclassical gravity. Its validity is conjectural, but we are able to extract an interesting nongravitational limit. Our proposal implies Wall's ant conjecture on the minimum energy of a completion of a quantum field theory state on a half-space. It further constrains the properties of the minimum energy state; for example, the minimum completion energy must be localized as a shock at the cut. We verify that the predicted properties hold in a recent explicit construction of Ceyhan and Faulkner, which proves our conjecture in the nongravitational limit.

A study on slip activation for a coarse-grained and single crystalline CoCrNi medium entropy alloy

The new CoCrNi medium entropy alloy (MEA) has emerged to be one of the most promising systems which provide extraordinary ductility and strength at cryogenic temperatures. In this study, utilizing both polycrystalline and single crystal specimens, as well as advanced optical strain measurements, the deformation mechanisms dictating the mechanical behavior at the onset of plasticity were detected and precisely quantified. Independent of deformation temperature, the accumulation of permanent strains at the microstructural level was attributed to plastic slip at the onset of yielding and at low levels of deformation (<10%). The resolved shear stress for slip activation was measured to be 78 MPa at 298 K and between 140 MPa and 160 MPa at 77 K. These unique measurements were used to provide an estimate of the temperature dependent resolved shear stress for slip at temperatures ranging from 77 K to 575 K. In addition, strain heterogeneities at the grain-scale were measured to study the nucleation of slip at the micro-scale in polycrystalline specimens. In summary, the present study aims to quantitatively assess the accumulation of plastic strain and reveal the underlying deformation mechanisms (i.e., slip or/twinning) leading to the buildup of plastic strains at the microstructural level.

Fitness Gain of Individually Sensed Information by Cells

Mutual information and its causal variant, directed information, have been widely used to quantitatively characterize the performance of biological sensing and information transduction. However, once coupled with selection in response to decision-making, the sensing signal could have more or less evolutionary value than its mutual or directed information. In this work, we show that an individually sensed signal always has a better fitness value, on average, than its mutual or directed information. The fitness gain, which satisfies fluctuation relations (FRs), is attributed to the selection of organisms in a population that obtain a better sensing signal by chance. A new quantity, similar to the coarse-grained entropy production in information thermodynamics, is introduced to quantify the total fitness gain from individual sensing, which also satisfies FRs. Using this quantity, the optimizing fitness gain of individual sensing is shown to be related to fidelity allocations for individual environmental histories. Our results are supplemented by numerical verifications of FRs, and a discussion on how this problem is linked to information encoding and decoding.

Soft mode and interior operator in the Hayden-Preskill thought experiment

We study the smoothness of the black hole horizon in the Hayden-Preskill thought experiment by using two particular toy models based on variants of Haar random unitary. The first toy model corresponds to the case where the coarse-grained entropy of a black hole is larger than its entanglement entropy. We find that, while the outgoing mode and the remaining black hole are entangled, the Hayden-Preskill recovery cannot be performed. The second toy model corresponds to the case where the system consists of low energy soft modes and high energy heavy modes. We find that the Hayden-Preskill recovery protocol can be carried out via soft modes whereas heavy modes give rise to classical correlations between the outgoing mode and the remaining black hole. We also point out that the procedure of constructing the interior partners of the outgoing soft mode operators can be interpreted as the Hayden-Preskill recovery, and as such, the known recovery protocol enables us to explicitly write down the interior operators. Hence, while the infalling mode needs to be described jointly by the remaining black hole and the early radiation in our toy model, adding a few extra qubits from the early radiation is sufficient to reconstruct the interior operators.

Entropy production for coarse-grained dynamics

Systems out of equilibrium exhibit a net production of entropy. We study the dynamics of a stochastic system represented by a Master equation (ME) that can be modeled by a Fokker–Planck equation in a coarse-grained, mesoscopic description. We show that the corresponding coarse-grained entropy production contains information on microscopic currents that are not captured by the Fokker–Planck equation and thus cannot be deduced from it. We study a discrete-state and a continuous-state system, deriving in both the cases an analytical expression for the coarse-graining corrections to the entropy production. This result elucidates the limits in which there is no loss of information in passing from a ME to a Fokker–Planck equation describing the same system. Our results are amenable of experimental verification, which could help to infer some information about the underlying microscopic processes.

Relaxation of one-dimensional collisionless gravitating systems

In an effort to better understand collisionless relaxation processes in gravitational systems, we investigate one-dimensional models. Taking advantage of a Hermite-Legendre expansion of relevant distribution functions, we present analytical and numerical behaviors of Maxwell-Boltzmann entropy. In particular, we modestly perturb systems about a separable-solution equilibrium and observe their collisionless evolution to a steady state. We verify the time-independence of fine-grained entropy in these systems before turning our attention to the behavior of coarse-grained entropy. We also verify that there is no analogue to the collisional H-theorem for these systems. Competing terms in the second-order coarse-grained entropy make it impossible to guarantee continuously increasing entropy. However, over dynamical time-scales the coarse-grained entropy generally increases, with small oscillations occurring. The lack of substantive differences between the entropies in test-particle and self-gravitating cases suggests that phase mixing, rather than violent relaxation associated with potential changes, more significantly drives the coarse-grained entropy evolution. The effects of violent relaxation can be better quantified through analysis of energy distributions rather than phase-space distributions.

High-cycle fatigue and tensile deformation behaviors of coarse-grained equiatomic CoCrFeMnNi high entropy alloy and unexpected hardening behavior during cyclic loading

High entropy alloy (HEA), a new class of materials, has received attention as a substance that can potentially replace conventional alloys. Equiatomic CoCrFeMnNi HEA is an attractive material with excellent strength-ductility combination and corrosion resistance, and it achieves greater performance in low temperatures. This study investigated the HCF and tensile deformation behavior of equiatomic CoCrFeMnNi HEA. In order to suggest the possibility of the material's application in an as-homogenized state, coarse-grained (CG) equiatomic CoCrFeMnNi HEA was prepared. Microstructural observation measured an average grain size of 245.5 μm, and it was confirmed to have a face-centered cubic (FCC) random solid solution. A tensile test confirmed that the yield strength and tensile strength are 293.1 MPa and 625.6 MPa, respectively, and change in the work hardening rate according to deformation twin (DT) evolution during tensile deformation was observed. A high-cycle fatigue results shows fatigue strength of 280 MPa, which is close to its yield strength, and this confirmed that the material has outstanding high-cycle fatigue properties considering its yield strength. DT is uniquely formed at cycle loading with a stress level lower than the critical twinning stress (σT), and this can improve yield strength by approximately 95% and tensile strength by approximately 17%. Based on the above findings, this study discussed the role of DTs which affect the high-cycle fatigue and deformation behavior of coarse-grained HEA.

Coarse graining holographic black holes

We expand our recent work on the outer entropy, a holographic coarse-grained entropy defined by maximizing the boundary entropy while fixing the classical bulk data outside some surface. When the surface is marginally trapped and satisfies certain “minimar” conditions, we prove that the outer entropy is exactly equal to a quarter the area (while for other classes of surfaces, the area gives an upper or lower bound). We explicitly construct the entropy-maximizing interior of a minimar surface, and show that it satisfies the appropriate junction conditions. This provides a statistical explanation for the area-increase law for spacelike holographic screens foliated by minimar surfaces. Our construction also provides an interpretation of the area for a class of non-minimal extremal surfaces.On the boundary side, we define an increasing simple entropy by maximizing the entropy subject to a set of “simple experiments” performed after some time. We show (to all orders in perturbation theory around equilibrium) that the simple entropy is the boundary dual to our bulk construction.

Outer entropy and quasilocal energy

We define the coarse-grained entropy of a `normal' surface $\sigma$, i.e., a surface that is neither trapped nor antitrapped. Following Engelhardt and Wall, the entropy is defined in terms of the area of an auxiliary extremal surface. This area is maximized over all auxiliary geometries that can be constructed in the interior of $\sigma$, while holding fixed the spatial exterior (the outer wedge). We argue that the area is maximized when the stress tensor in the auxiliary geometry vanishes, and we develop a formalism for computing it under this assumption. The coarse-grained entropy can be interpreted as a quasilocal energy of $\sigma$. This energy possesses desirable properties such as positivity and monotonicity, which derive directly from its information-theoretic definition.

Systematic Coarse-Grained Lipid Force Fields with Semiexplicit Solvation via Virtual Sites.

Despite the central role of lipids in many biophysical functions, the molecular mechanisms that dictate macroscopic lipid behavior remain elusive to both experimental and computational approaches. As such, there has been much interest in the development of low-resolution, implicit-solvent coarse-grained (CG) models to dynamically simulate biologically relevant spatiotemporal scales with molecular fidelity. However, in the absence of solvent, a key challenge for CG models is to faithfully emulate solvent-mediated forces, which include both hydrophilic and hydrophobic interactions that drive lipid aggregation and self-assembly. In this work, we provide a new methodological framework to incorporate semiexplicit solvent effects through the use of virtual CG particles, which represent structural features of the solvent-lipid interface. To do so, we leverage two systematic coarse-graining approaches, multiscale coarse-graining (MS-CG) and relative entropy minimization (REM), in a hybrid fashion to construct our virtual-site CG (VCG) models. As a proof-of-concept, we focus our efforts on two lipid species, 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) and 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), which adopt a liquid-disordered and gel phase, respectively, at room temperature. Through our analysis, we also present, to our knowledge, the first direct comparison between the MS-CG and REM methods for a complex biomolecule and highlight each of their strengths and weaknesses. We further demonstrate that VCG models recapitulate the rich biophysics of lipids, which enable self-assembly, morphological diversity, and multiple phases. Our findings suggest that the VCG framework is a powerful approach for investigation into macromolecular biophysics.

Quantum coarse-grained entropy and thermodynamics

We extend classical coarse-grained entropy, commonly used in many branches of physics, to the quantum realm. We find two coarse-grainings, one using measurements of local particle numbers and then total energy, and the second using local energy measurements, which lead to an entropy that is defined outside of equilibrium, is in accord with the thermodynamic entropy for equilibrium systems, and reaches the thermodynamic entropy in the long-time limit, even in genuinely isolated quantum systems. This answers the long-standing conceptual problem, as to which entropy is relevant for the formulation of the second thermodynamic law in closed quantum systems. This entropy could be in principle measured, especially now that experiments on such systems are becoming feasible.

Coarse Graining, Nonmaximal Entropy, and Power Laws.

We show that coarse graining produces significant and predictable effects on the entropy of states of equilibrium when the scale of coarse graining becomes comparable to that of density fluctuations. We demonstrate that a coarse-grained entropy typically evolves toward a state of effective equilibrium with a lower value than that of the state of maximum entropy theoretically possible. The finer the coarse graining, the greater the drop in effective entropy, and the more relevant the fluctuations around that. Fundamental considerations allow us to derive a remarkable power law that relates coarse graining to the effective entropy gap. Another power law is found that precisely relates the noise range of effective entropy fluctuations to coarse graining. We test both power laws with numerical simulations based on a well-studied two-dimensional lattice gas model. As expected, the effects of these power laws diminish as our description approaches a macroscopic level, eventually disappearing in the thermodynamic limit, where the maximum entropy principle is reasserted.