arXiv subjects

Ralph V. Chamberlin

Publications and source records attributed to Ralph V. Chamberlin.

At least 19 recordsLinked to original sources

arXiv2026

Nanothermodynamics: stable thermal equilibrium and nanoscale fluctuations

Nanothermodynamics describes the process where large systems subdivide into equilibrium distributions of small subsystems. A key ingredient is Hill's subdivision potential (E) that ensures adherence to the 1st and 2nd laws of thermodynamics in systems of any size. In this review and reassessment, it is emphasized that nanothermodynamics gives new insight into many measurements, theories, and simulations. Measurements establishing the need for nanothermodynamics show thermodynamic heterogeneity from multiple effective temperatures (T_i) inside most types of materials. One theoretical result that requires E=0 is the stable solution of Ising's original model for finite chains of interacting spins, a solution Ising could not have found 40 years before Hill's work. Another result is a novel solution to Gibbs' paradox that makes the entropy of the semiclassical ideal gas exactly extensive. Molecular dynamics simulations reveal how a standard fluctuation relation is modified when local degrees of freedom fluctuate faster than their coupling to the heat bath, consistent with the measured thermodynamic heterogeneity. Simulations of a Creutz-like model, comprised of Ising spins coupled to an explicit heat bath of Einstein oscillators, are used to study the 2nd law. It is found that maximizing the total entropy (S_t) requires an intrinsically irreversible step, providing a counterexample to the usual claim that statistical mechanics emerges from reversible dynamics. Furthermore, fluctuations of this model are best described by Einstein's reversal of Boltzmann's relation and the 2nd-law, not by recent fluctuation theorems.

cond-mat.stat-mech↗

arXiv2026

Excess energy fluctuations with applications to deviations from Debye's specific heat

Measured specific heats often exceed Debye's T^3-law, even in high-purity single crystals. Analogous excess energy fluctuations are found in molecular dynamics (MD) simulations of crystals with no defects. Here, a theory is developed for the fluctuations based on rapid modulation of the ground-state energy due to the motion of next-nearest-neighbor atoms. Crucially, the modulations must be adiabatic, with time- and phase-averaging before thermal averaging, consistent with evidence from the simulations and multiple experimental techniques showing that localized excitations are decoupled from the heat bath. Emergent nonextensivity is interpreted by analogy with anomalous diffusion. The theory modifies the standard relation between energy fluctuations and specific heat, giving good agreement with the simulations and new insight into many measurements. The theory may also provide a basis for understanding excess specific heat in amorphous materials and anomalous noise in quantum devices.

cond-mat.stat-mech↗

arXiv2024

Nanothermodynamics: There's plenty of room on the inside

Nanothermodynamics provides the theoretical foundation for understanding stable distributions of statistically independent subsystems inside larger systems. In this review it is emphasized that adapting ideas from nanothermodynamics to simplistic models improves agreement with the measured properties of many materials. Examples include non-classical critical scaling near ferromagnetic transitions, thermal and dynamic behavior near liquid-glass transitions, and the 1/f-like noise in metal films and qubits. A key feature in several models is to allow separate time steps for distinct conservation laws: one type of step conserves energy and the other conserves momentum (e.g. dipole alignment). This "orthogonal dynamics" explains how the relaxation of a single parameter can exhibit multiple responses such as primary, secondary, and microscopic peaks in the dielectric loss of supercooled liquids, and the crossover in thermal fluctuations from Johnson-Nyquist (white) noise at high frequencies to 1/f-like noise at low frequencies. Nanothermodynamics also provides new insight into three basic questions. First, it gives a novel solution to Gibbs' paradox for the entropy of the semi-classical ideal gas. Second, it yields the stable equilibrium of Ising's original model for finite-sized chains of interacting binary degrees of freedom ("spins"). Third, it confronts Loschmidt's paradox for the arrow of time, showing that an intrinsically irreversible step is required for maximum entropy and the second law of thermodynamics, not only in the thermodynamic limit but also in systems as small as N=2 particles

cond-mat.stat-mech↗

arXiv2024

On "Nanoscale thermodynamics needs the concept of a disjoining chemical potential", by W. Dong, Nat. Comm. 10, 1038 (2023)

In a 2023 Nature Communication, Dong claims that a disjoining chemical potential should be used instead of "Hill's nanothermodynamics" because Hill's subdivision potential "remains elusive to relate...to any experimental observables." Here, I remind readers of the 2000 Letter to Nature where Hill's ideas were adapted to provide improved agreement with the non-classical critical scaling measured in ferromagnetic materials, and where the term "nanothermodynamics" first appeared. I also suggest how adding nanothermodynamics can improve Dong's results for measurements on real systems. Because Dong's claim comes from his failure to recognize the relevance of Hill's work in the original Letter, and in dozens of other publications, I propose how the term nanothermodynamics might be clarified to avoid future confusion.

cond-mat.stat-mech↗

arXiv2023

Small and simple systems that favor the arrow of time

The 2nd law of thermodynamics yields an irreversible increase in entropy until thermal equilibrium is achieved. This irreversible increase is often assumed to require large and complex systems to emerge from the reversible microscopic laws of physics. We test this assumption using simulations and theory of a 1D ring of N Ising spins coupled to an explicit heat bath of N Einstein oscillators. The exact entropy of the spins and bath can be calculated for any N, with dynamics that is readily altered from reversible to irreversible. We find thermal equilibrium behavior in the thermodynamic limit, and in systems as small as N=2, but only if the microscopic dynamics is intrinsically irreversible.

cond-mat.stat-mech↗

arXiv2022

An Ising model for the thermal and dynamic properties of supercooled liquids and the glass transition

We describe the behavior of an Ising model with orthogonal dynamics, where changes in energy and changes in alignment never occur during the same Monte Carlo (MC) step. This orthogonal Ising model (OIM) allows conservation of energy and conservation of momentum to proceed independently, on their own preferred time scales. MC simulations of the OIM mimic more than twenty distinctive characteristics that are commonly found above and below the glass temperature, Tg. Examples include a specific heat that has hysteresis around Tg, out-of-phase loss that exhibits primary and secondary peaks, super-Arrhenius T dependence for the alpha response time, and fragilities that increase with increasing system size (N). Mean-field theory for energy fluctuations in the OIM yields a novel expression for the super-Arrhenius divergence. Because this divergence is reminiscent of the Vogel-Fulcher-Tammann (VFT) law squared, we call it the VFT2 law. A modified Stickel plot, which linearizes the VFT2 law, gives qualitatively consistent agreement with measurements of primary response (from the literature) on five glass-forming liquids. Such agreement with the OIM suggests that several basic features govern supercooled liquids. The freezing of a liquid into a glass involves an underlying 2nd-order transition that is broadened by finite-size effects. The VFT2 law comes from energy fluctuations that enhance the pathways through an entropy bottleneck, not activation over an energy barrier. Primary response times vary exponentially with inverse N, consistent with the distribution of relaxation times deduced from measurements. System sizes found via the T dependence of the primary response are similar to sizes of independently relaxing regions measured by nuclear magnetic resonance for simple-molecule glass-forming liquids. The OIM provides a broad foundation for more-detailed models of liquid-glass behavior.

cond-mat.soft↗

arXiv2021

Multiscale Thermodynamics: Energy, Entropy, and Symmetry from Atoms to Bulk Behavior

Here we investigate how local properties of particles in a thermal bath influence the thermodynamics of the bath. We utilize nanothermodynamics, based on two postulates: that small systems can be treated self-consistently by coupling to an ensemble of similarly small systems, and that a large ensemble of small systems forms its own thermodynamic bath. We adapt these ideas to study how a large system may subdivide into an ensemble of smaller subsystems, causing internal heterogeneity across multiple size scales. For the semi-classical ideal gas, maximum entropy favors subdividing a large system of atoms into regions of variable size. The mechanism of region formation could come from quantum exchange that makes atoms in each region indistinguishable, while decoherence between regions allows atoms in separate regions to be distinguishable by location. Combining regions reduces the total entropy, as expected when distinguishable particles become indistinguishable, and as required by theorems for sub-additive entropy. Combining large volumes of small regions gives the entropy of mixing for a semi-classical ideal gas, resolving Gibbs paradox without invoking quantum symmetry for distant atoms. Other models we study are based on Ising-like spins in 1-D. We find similarity in the properties of a two-state model in the nanocanonical ensemble and a three-state model in the canonical ensemble. Thus, emergent phenomena may alter the thermal behavior of microscopic models, and the correct ensemble is necessary for fully-accurate predictions. We add a nonlinear correction to Boltzmann's factor in simulations of the Ising-like spins to imitate the dynamics of spin exchange on intermediate lengths, yielding the statistics of indistinguishable states. These simulations exhibit 1/f-like noise at low frequencies (f), and white noise at higher f, similar to the thermal fluctuations found in many materials.

cond-mat.stat-mech↗

arXiv2020

An introduction to nanothermodynamics: Thermal equilibrium for heterogeneous and finite-sized systems

The theory of small-system thermodynamics was originally developed to extend the laws of thermodynamics to length scales of nanometers. Here we review this "nanothermodynamics," and stress how it also applies to large systems that subdivide into a heterogeneous distribution of internal subsystems that we call "regions." We emphasize that the true thermal equilibrium of most systems often requires that these regions are in the fully-open generalized ensemble, with a distribution of region sizes that is not externally constrained, which we call the "nanocanonical" ensemble. We focus on how nanothermodynamics impacts the statistical mechanics of specific models. One example is an ideal gas of indistinguishable atoms in a large volume that subdivides into an ensemble of small regions of variable volume, with separate regions containing atoms that are distinguishable from those in other regions. Combining such subdivided regions yields the correct entropy of mixing, avoiding Gibbs paradox without resorting to macroscopic quantum symmetry for semi-classical particles. Other models are based on Ising-like spins (binary degrees of freedom), which are solved analytically in one-dimension, making them suitable examples for introductory courses in statistical physics. A key result is to quantify the net increase in entropy when large systems subdivide into small regions of variable size. Another result is to show similarity in the equilibrium properties of a two-state model in the nanocanonical ensemble and a three-state model in the canonical ensemble. Thus, emergent phenomena may alter the thermal behavior of microscopic models, and the correct ensemble is necessary for accurate predictions.

cond-mat.stat-mech↗

arXiv2019

Energy localization and excess fluctuations from long-range interactions in equilibrium molecular dynamics

Molecular Dynamics (MD) simulations of standard systems of interacting particles ("atoms") give excellent agreement with the equipartition theorem for the average energy, but we find that these simulations exhibit finite-size effects in the dynamics that cause local fluctuations in energy to deviate significantly from the analogous energy fluctuation relation (EFR). The main conclusion of our study is that systems separated into nanometer-sized "blocks" inside much larger simulations exhibit excess fluctuations in potential energy (pe) that diverge inversely proportional to T in a manner that is strongly dependent on the range of interaction. Specifically, at low T with long-range interactions pe fluctuations exceed the EFR by at least an order of magnitude, dropping abruptly to below the EFR when interactions include only 1st-neighbor atoms. A simplistic model that includes 2nd-neighbor interactions matches the behavior of the excess pe fluctuations, but only if the 2nd-neighbor terms are not included in Boltzmann's factor, attributable to energy localization due to anharmonic effects. Characterizing energy correlations as a function of time and distance reveals that excess pe fluctuations in a block coincide with negative pe correlations between neighboring blocks, whereas reduced pe fluctuations coincide with positive pe correlations. Indeed, anomalous pe fluctuations in small systems at low T can be quantified by using the net energy in Boltzmann's factor that includes the pe from a surrounding shell of similarly small systems, or equivalently an effective local temperature. Our analysis elucidates the source of non-Boltzmann fluctuations, and the need to include mesoscopic thermal effects from the local environment for a consistent theoretical description of the equilibrium fluctuations in MD simulations of standard models with long-range interactions.

physics.comp-ph↗

arXiv2016

Lindbladian operators, von Neumann entropy and energy conservation in time-dependent quantum open systems

The Lindblad equation is widely employed in studies of Markovian quantum open systems. Here, firstly, a simple result is presented on the time evolution of the non Neumann entropy under the Lindblad equation, which enables one to examine if the entropy increases/decreases. Then, secondly, the following question is posed: In a quantum open system with a time-dependent Hamiltonian, what is the corresponding Lindblad equation for the quantum state that keeps the internal energy of the system constant in time? This issue is of importance in realizing quasi-stationary states of open systems such as quantum circuits and batteries. As an illustrative example, the time-dependent harmonic oscillator is analyzed. The Lindbladian operator is uniquely determined with the help of a Lie-algebraic structure, and the time derivative of the von Neumann entropy is shown to be nonnegative depending on the time-dependence of the Hamiltonian.

quant-ph↗

arXiv2016

Fluctuation theorems and 1/f noise from a simple matrix

Here we present a model for a small system combined with an explicit entropy bath that is comparably small. The dynamics of the model is defined by a simple matrix, M. Each row of M corresponds to a macrostate of the system, e.g. net alignment, while the elements in the row represent microstates. The constant number of elements in each row ensures constant entropy, which allows reversible fluctuations, similar to information theory where a constant number of bits allows reversible computations. Many elements in M come from the microstates of the system, but many others come from the bath. Bypassing the bath states yields fluctuations that exhibit standard white noise; whereas with bath states the power spectral density varies as S(f)~1/f over a wide range of frequencies, f. Thus, the explicit entropy bath is the mechanism of 1/f noise in this model. Both forms of the model match Crooks' fluctuation theorem exactly, indicating that the theorem applies not only to infinite reservoirs, but also to finite-sized baths. The model is used to analyze measurements of 1/f-like noise from a sub-micron tunnel junction.

cond-mat.stat-mech↗

arXiv2015

The Big World of Nanothermodynamics

Nanothermodynamics extends standard thermodynamics to facilitate finite-size effects on the scale of nanometers. A key ingredient is Hill's subdivision potential that accommodates the non-extensive energy of independent small systems, similar to how Gibbs' chemical potential accommodates distinct particles. Nanothermodynamics is essential for characterizing the thermal equilibrium distribution of independently relaxing regions inside bulk samples, as is found for the primary response of most materials using various experimental techniques. The subdivision potential ensures strict adherence to the laws of thermodynamics: total energy is conserved by including an instantaneous contribution from the entropy of local configurations, and total entropy remains maximized by coupling to a thermal bath. A unique feature of nanothermodynamics is the completely-open nanocanonical ensemble. Another feature is that particles within each region become statistically indistinguishable, which avoids non-extensive entropy, and mimics quantum-mechanical behavior. Applied to mean-field theory, nanothermodynamics gives a heterogeneous distribution of regions that yields stretched-exponential relaxation and super-Arrhenius activation. Applied to Monte Carlo simulations, there is a nonlinear correction to Boltzmann's factor that improves agreement between the Ising model and measured non-classical critical scaling in magnetic materials. Nanothermodynamics also provides a fundamental mechanism for the 1/f noise found in many materials.

cond-mat.stat-mech↗

arXiv2014

1/f Noise from the Laws of Thermodynamics for Finite-Size Fluctuations

Computer simulations of the Ising model exhibit white noise if thermal fluctuations are governed by Boltzmann's factor alone; whereas we find that the same model exhibits 1/f noise if Boltzmann's factor is extended to include local alignment entropy to all orders. We show that this nonlinear correction maintains maximum entropy during equilibrium fluctuations. Indeed, as with the usual resolution of Gibbs' paradox that avoids net entropy reduction during reversible processes, the correction yields the statistics of indistinguishable particles. The correction also ensures conservation of energy if an instantaneous contribution from local entropy is included. Thus, a common mechanism for 1/f noise comes from assuming that finite-size fluctuations strictly obey the laws of thermodynamics, even in small parts of a large system. Empirical evidence for the model comes from its ability to match the measured temperature dependence of the spectral-density exponents in several metals, and to show non-Gaussian fluctuations characteristic of nanoscale systems.

cond-mat.mes-hall↗

arXiv2013

Modified Bose-Einstein and Fermi-Dirac statistics if excitations are localized on an intermediate length scale: Application to non-Debye specific heat

Disordered systems show deviations from the standard Debye theory of specific heat at low temperatures. These deviations are often attributed to two-level systems of uncertain origin. We find that a source of excess specific heat comes from correlations between quanta of energy if phonon-like excitations are localized on an intermediate length scale. We use simulations of a simplified Creutz model for a system of Ising-like spins coupled to a thermal bath of Einstein-like oscillators. One feature of this model is that energy is quantized in both the system and its bath, ensuring conservation of energy at every step. Another feature is that the exact entropies of both the system and its bath are known at every step, so that their temperatures can be determined independently. We find that there is a mismatch in canonical temperature between the system and its bath. In addition to the usual finite-size effects in the Bose-Einstein and Fermi-Dirac distributions, if excitations in the heat bath are localized on an intermediate length scale, this mismatch is independent of system size up to at least 10^6 particles. We use a model for correlations between quanta of energy to adjust the statistical distributions and yield a thermodynamically consistent temperature. The model includes a chemical potential for units of energy, as is often used for other types of particles that are quantized and conserved. Experimental evidence for this model comes from its ability to characterize the excess specific heat of imperfect crystals at low temperatures.

cond-mat.stat-mech↗

arXiv2009

Beyond the Boltzmann factor for corrections to scaling in ferromagnetic materials and critical fluids

The Boltzmann factor comes from the linear change in entropy of an infinite heat bath during a local fluctuation; small systems have significant nonlinear terms. We present theoretical arguments, experimental data, and Monte-Carlo simulations indicating that nonlinear terms may also occur when a particle interacts directly with a finite number of neighboring particles, forming a local region that fluctuates independent of the infinite bath. A possible mechanism comes from the net force necessary to change the state of a particle while conserving local momentum. These finite-sized local regions yield nonlinear fluctuation constraints, beyond the Boltzmann factor. One such fluctuation constraint applied to simulations of the Ising model lowers the energy, makes the entropy extensive, and greatly improves agreement with the corrections to scaling measured in ferromagnetic materials and critical fluids.

cond-mat.stat-mech↗

arXiv2009

Fluctuation-theory constraint for extensive entropy in Monte-Carlo simulations

The entropy per particle in most Monte-Carlo simulations is size dependent due to correlated energy fluctuations. Guided by nanothermodynamics, we find a constraint for the Ising model that enhances the fluctuations and lowers the free energy, while making the entropy homogeneous, additive, and extensive. Although the average interaction energy becomes size dependent, the resulting distribution of energies provides a mechanism for the heterogeneity found in the dynamics of many materials.

cond-mat.stat-mech↗

arXiv2007

Comment on the history of the stretched exponential function

The original article of Rudolf Kohlrausch (1854) on stretched exponentials and their application to describe relaxation phenomena has been often misquoted in the literature after its rediscovery around 1984. We discuss here this fact and attempt to set the record straight.

physics.hist-ph↗

arXiv2006

A free-energy landscape picture and Landau theory for the dynamics of disordered materials

Landau's theory of phase transitions is adapted to treat independently relaxing regions in complex systems using nanothermodynamics. The order parameter we use governs the thermal fluctuations, not a specific static structure. We find that the entropy term dominates the thermal behavior, as is reasonable for disordered systems. Consequently, the thermal equilibrium occurs at the internal-energy maximum, so that the minima in a potential-energy landscape have negligible influence on the dynamics. Instead the dynamics involves normal thermal fluctuations about the free-energy minimum, with a time scale that is governed by the internal-energy maximum. The temperature dependence of the fluctuations yields VTF-like relaxation rates and approximate time-temperature superposition, consistent with the WLF procedure for analyzing the dynamics of complex fluids; while the size dependence of the fluctuations provides an explanation for the distribution of relaxation times and heterogeneity that are found in glass-forming liquids, thus providing a unified picture for several features in the dynamics of disordered materials.

cond-mat.dis-nn↗