Thermodynamics Of Small Systems Research Articles

Simulations evidencing two surface tensions for fluids confined in nanopores

As first recognized by Hill in 1960′s, one character distinguishing the thermodynamics of small systems from the macroscopic one is that some intensive variables are no longer defined uniquely for small systems. For example, the differential chemical potential, defined as the derivative of a thermodynamic potential with respect to particle number, is no longer equal to the integral one, given by the ratio of Gibbs free energy to the particle number. The concept of differential and integral surface tensions has been introduced recently to account for the increasing surface contribution to thermodynamic potentials when a system shrinks down in size. Simulations constitute a powerful tool for testing new concepts. The present work provides the simulation evidence for distinct differential and integral surface tensions. Our results point out some useful directions for future experimental investigations to check the general validity of the concept of differential and integral surface tensions.

Quantum thermodynamics of small systems: The anyonic otto engine

Recent advances in applying thermodynamic ideas to quantum systems have raised the novel prospect of using non-thermal, non-classical sources of energy, of purely quantum origin, like quantum statistics, to extract mechanical work in macroscopic quantum systems like Bose–Einstein condensates. On the other hand, thermodynamic ideas have also been applied to small systems like single molecules and quantum dots. In this paper, we study the quantum thermodynamics of small systems of anyons, with specific emphasis on the quantum Otto engine which uses, as its working medium, just one or two anyons. Formulae are derived for the efficiency of the Otto engine as a function of the statistics parameter.

Gas–droplet–liquid transitions and fluctuations in soft nano-confinement

One permanent characteristic of the thermodynamics of small systems is environment-dependence, also known as ensemble-dependence. Fluid molecules in soft (deformable) nano-confinement offer a special ensemble that acts as a bridge between classical isobaric (NPT) and isochoric (NVT) ensembles. Here, we discuss the gas–liquid transition taking place in a soft nano-confinement where the cell volume is not fixed but changes when the system pressure is changed. The free energy of the system is calculated as a function of the size of the liquid droplet that appears in the gas phase. We discuss how the phase behavior changes when the condition of the confinement changes from rigid confinement to very soft confinement. For the simple fluid model studied, the coexistence and critical phase behaviors are found to be uniquely determined by αK (αK is the dimensionless elasticity constant of the wall of confined space and is proportional to its ability to resist deformation), and the confinement with moderate softness exhibits richer phase behavior. We then study the fluctuations of pressure, volume, and droplet size for fluid in soft confined spaces, which is again closely related to the wall softness. Under moderate softness, large fluctuations in both fluid pressure and volume are seen in the transition region where fluid pressure increases with volume expansion, accompanied by the strengthened fluctuation of droplet size.

Microcanonical Thermodynamics of Small Ideal Gas Systems

We consider the thermal, mechanical, and chemical contact of two subsystems composed of ideal gases, both of which are not in the thermodynamic limit. After contact, the combined system is isolated, and the entropy is determined through the use of its standard connection to the phase space density (PSD), where only those microstates at a given energy value are counted. The various intensive properties of these small systems that follow from a derivative of the PSD, such as the temperature, pressure, and chemical potential (evaluated via a backward difference), while equal when the two subsystems are in equilibrium are nevertheless found not to behave in accordance with what is expected from macroscopic thermodynamics. Instead, it is the entropy, defined from its connection to the PSD, that still controls the behavior of these small (nonextensive) systems. We also analyze the contact of these two subsystems utilizing an alternative entropy definition, through its proposed connection to the phase space volume (PSV), where all microstates at or below a given energy value are counted. We show that certain key properties of these small systems obtained with the PSV either do not become equal or do not consistently describe the two subsystems when in contact, suggesting that the PSV should not be used for analyzing the behavior of small isolated systems.

Nanoscale thermodynamics needs the concept of a disjoining chemical potential

Disjoining pressure was discovered by Derjaguin in 1930’s, which describes the difference between the pressure of a strongly confined fluid and the corresponding one in a bulk phase. It has been revealed recently that the disjoining pressure is at the origin of distinct differential and integral surface tensions for strongly confined fluids. Here we show how the twin concept, disjoining chemical potential, arises in a reminiscent way although it comes out eighty years later. This twin concept advances our understanding of nanoscale thermodynamics. Ensemble-dependence (or environment-dependence) is one hallmark of thermodynamics of small systems. We show that integral surface tension is ensemble-dependent while differential surface tension is not. Moreover, two generalized Gibbs-Duhem equations involving integral surface tensions are derived, as well as two additional adsorption equations relating surface tensions to adsorption-induced strains. All the results obtained in this work further evidence that an approach alternative of Hill’s nanothermodynamics is possible, by extending Gibbs surface thermodynamics instead of resorting to Hill’s replica trick. Moreover, we find a compression-expansion hysteresis without any underlying phase transition.

On small local equilibrium systems

Abstract Even for large nonequilibrium systems, local equilibrium subsystems in the presence of strong inhomogeneities may be very small. Such situations typically arise either in the presence of large gradients of temperature, velocity or pressure, or in transition zones between different phases. For small thermodynamic systems, the Euler equation of macroscopic thermodynamics does not hold. One less equation implies one additional degree of freedom, which is the hallmark of small thermodynamic systems. I would like to offer some remarks on the description and role of small local equilibrium subsystems in nonequilibrium thermodynamics.

A Statistical Perspective on Microsolvation.

The lack of a procedure to determine equilibrium thermodynamic properties of a small system interacting with a bath is frequently seen as a weakness of conventional statistical mechanics. A typical example for such a small system is a solute surrounded by an explicit solvation shell. One way to approach this problem is to enclose the small system of interest in a large bath of explicit solvent molecules, considerably larger than the system itself. The explicit inclusion of the solvent degrees of freedom is obviously limited by the available computational resources. A potential remedy to this problem is a microsolvation approach where only a few explicit solvent molecules are considered and surrounded by an implicit solvent bath. Still, the sampling of the solvent degrees of freedom is challenging with conventional grand canonical Monte Carlo methods, since no single chemical potential for the solvent molecules can be defined in the realm of small-system thermodynamics. In this work, a statistical thermodynamic model based on the grand canonical ensemble is proposed that avoids the conventional system size limitations and accurately characterizes the properties of the system of interest subject to the thermodynamic constraints of the bath. We extend an existing microsolvation approach to a generalized multibath "microstatistical" model and show that the previously derived approaches result as a limit of our model. The framework described here is universal and we validate our method numerically for a Lennard-Jones model fluid.

Defining the pressures of a fluid in a nanoporous, heterogeneous medium

We describe the thermodynamic state of a single-phase fluid confined to a porous medium with Hill’s thermodynamics of small systems, also known as nanothermodynamics. This way of defining small system thermodynamics, with a separate set of control variables, may be useful for the study of transport in non-deformable porous media, where presently no consensus exists on pressure computations. For a confined fluid, we observe that there are two pressures, the integral and the differential pressures. We use molecular simulations to investigate and confirm the nanothermodynamic relations for a representative elementary volume (REV). For a model system of a single-phase fluid in a face-centered cubic lattice of solid spheres of varying porosity, we calculate the fluid density, fluid-solid surface tension, replica energy, integral pressure, entropy, and internal energy.

Quasi-static Decomposition and the Gibbs Factorial in Small Thermodynamic Systems

For a classical system consisting of N-interacting identical particles in contact with a heat bath, we define the free energy from thermodynamic relations in equilibrium statistical mechanics. Concretely, the temperature dependence of the free energy is determined from the Gibbs-Helmholtz relation, and its volume dependence is determined from the condition that the quasi-static work in a volume change is equal to the free energy change. Now, we argue the free energy difference in a quasi-static decomposition of small thermodynamic systems. We can then determine the N dependence of the free energy, which includes the Gibbs factorial N! in addition to the phase space integration of the Gibbs–Boltzmann factor.

Nonequilibrium Control of Thermal and Mechanical Changes in a Levitated System.

Fluctuation theorems are fundamental extensions of the second law of thermodynamics for small nonequilibrium systems. While work and heat are equally important forms of energy exchange, fluctuation relations have not been experimentally assessed for the generic situation of simultaneous mechanical and thermal changes. Thermal driving is indeed generally slow and more difficult to realize than mechanical driving. Here, we use feedback cooling techniques to implement fast and controlled temperature variations of an underdamped levitated microparticle that are 1 order of magnitude faster than the equilibration time. Combining mechanical and thermal control, we verify the validity of a fluctuation theorem that accounts for both contributions, well beyond the range of linear response theory. Our results allow the investigation of general far-from-equilibrium processes in microscopic systems that involve fast mechanical and thermal changes at the same time.

Small size effects in open and closed systems: What can we learn from ideal gases about systems with interacting particles?

Small systems have higher surface area-to-volume ratios than macroscopic systems. The thermodynamics of small systems therefore deviates from the description of classical thermodynamics. One consequence of this is that properties of small systems can be dependent on the system's ensemble. By comparing the properties in grand canonical (open) and canonical (closed) systems, we investigate how a small number of particles can induce an ensemble dependence. Emphasis is placed on the insight that can be gained by investigating ideal gases. The ensemble equivalence of small ideal gas systems is investigated by deriving the properties analytically, while the ensemble equivalence of small systems with particles interacting via the Lennard-Jones or the Weeks-Chandler-Andersen potential is investigated through Monte Carlo simulations. For all the investigated small systems, we find clear differences between the properties in open and closed systems. For systems with interacting particles, the difference between the pressure contribution to the internal energy, and the difference between the chemical potential contribution to the internal energy, are both increasing with the number density. The difference in chemical potential is, with the exception of the density dependence, qualitatively described by the analytic formula derived for an ideal gas system. The difference in pressure, however, is not captured by the ideal gas model. For the difference between the properties in the open and closed systems, the response of increasing the particles' excluded volume is similar to the response of increasing the repulsive forces on the system walls. This indicates that the magnitude of the difference between the properties in open and closed systems is related to the restricted movement of the particles in the system. The work presented in this paper gives insight into the mechanisms behind ensemble in-equivalence in small systems, and illustrates how a simple statistical mechanical model, such as the ideal gas, can be a useful tool in these investigations.

Fluctuation-Dissipation Theorems for Multiphase Flow in Porous Media.

A thermodynamic description of porous media must handle the size- and shape-dependence of media properties, in particular on the nano-scale. Such dependencies are typically due to the presence of immiscible phases, contact areas and contact lines. We propose a way to obtain average densities suitable for integration on the course-grained scale, by applying Hill’s thermodynamics of small systems to the subsystems of the medium. We argue that the average densities of the porous medium, when defined in a proper way, obey the Gibbs equation. All contributions are additive or weakly coupled. From the Gibbs equation and the balance equations, we then derive the entropy production in the standard way, for transport of multi-phase fluids in a non-deformable, porous medium exposed to differences in boundary pressures, temperatures, and chemical potentials. Linear relations between thermodynamic fluxes and forces follow for the control volume. Fluctuation-dissipation theorems are formulated for the first time, for the fluctuating contributions to fluxes in the porous medium. These give an added possibility for determination of the Onsager conductivity matrix for transport through porous media. Practical possibilities are discussed.

Fluctuating temperature outside superstatistics: Thermodynamics of small systems

The existence of fluctuations of temperature has been a somewhat controversial topic in thermodynamics but nowadays it is recognized that they must be taken into account in small, finite systems. Although for nonequilibrium steady states superstatistics is becoming the de facto framework for expressing such temperature fluctuations, some recent results put into question the idea of temperature as a phase space observable. In this work we present and explore the statistics that describes a part of an isolated system, small enough to have well-defined uncertainties in energy and temperature, but lacking a superstatistical description. These results motivate the use of the so-called fundamental temperature as an observable and may be relevant for the statistical description of small systems in physical chemistry.

Experimental Validation of Fully Quantum Fluctuation Theorems Using Dynamic Bayesian Networks.

Fluctuation theorems are fundamental extensions of the second law of thermodynamics for small systems. Their general validity arbitrarily far from equilibrium makes them invaluable in nonequilibrium physics. So far, experimental studies of quantum fluctuation relations do not account for quantum correlations and quantum coherence, two essential quantum properties. We here apply a novel dynamic Bayesian network approach to experimentally test detailed and integral fully quantum fluctuation theorems for heat exchange between two quantum-correlated thermal spins-1/2 in a nuclear magnetic resonance setup. We concretely verify individual integral fluctuation relations for quantum correlations and quantum coherence, as well as for the sum of all quantum contributions. We further investigate the thermodynamic cost of creating correlations and coherence.

Machine learning implicit solvation for molecular dynamics.

Accurate modeling of the solvent environment for biological molecules is crucial for computational biology and drug design. A popular approach to achieve long simulation time scales for large system sizes is to incorporate the effect of the solvent in a mean-field fashion with implicit solvent models. However, a challenge with existing implicit solvent models is that they often lack accuracy or certain physical properties compared to explicit solvent models as the many-body effects of the neglected solvent molecules are difficult to model as a mean field. Here, we leverage machine learning (ML) and multi-scale coarse graining (CG) in order to learn implicit solvent models that can approximate the energetic and thermodynamic properties of a given explicit solvent model with arbitrary accuracy, given enough training data. Following the previous ML-CG models CGnet and CGSchnet, we introduce ISSNet, a graph neural network, to model the implicit solvent potential of mean force. ISSNet can learn from explicit solvent simulation data and be readily applied to molecular dynamics simulations. We compare the solute conformational distributions under different solvation treatments for two peptide systems. The results indicate that ISSNet models can outperform widely used generalized Born and surface area models in reproducing the thermodynamics of small protein systems with respect to explicit solvent. The success of this novel method demonstrates the potential benefit of applying machine learning methods in accurate modeling of solvent effects for in silico research and biomedical applications.

Pseudogauge dependence of quantum fluctuations of the energy in a hot relativistic gas of fermions

Explicit expressions for quantum fluctuations of energy in subsystems of a hot relativistic gas of spin-$1/2$ particles are derived. The results depend on the form of the energy-momentum tensor used in the calculations, which is a feature described as pseudo-gauge dependence. However, for sufficiently large subsystems the results obtained in different pseudo-gauges converge and agree with the canonical-ensemble formula known from statistical physics. As different forms of the energy-momentum tensor of a gas are a priori equivalent, our finding suggests that the concept of quantum fluctuations of energy in very small thermodynamic systems is pseudo-gauge dependent. On the practical side, the results of our calculations determine a scale of coarse graining for which the choice of the pseudo-gauge becomes irrelevant.

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

Here, we investigate how the local properties of particles in a thermal bath may influence the thermodynamics of the bath, and consequently alter the statistical mechanics of subsystems that comprise the bath. We are guided by the theory of small-system thermodynamics, which is based on two primary postulates: that small systems can be treated self-consistently by coupling them to an ensemble of similarly small systems, and that a large ensemble of small systems forms its own thermodynamic bath. We adapt this “nanothermodynamics” to investigate 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 an ensemble of “regions” of variable size. The mechanism of region formation could come from quantum exchange symmetry that makes atoms in each region indistinguishable, while decoherence between regions allows atoms in separate regions to be distinguishable by their distinct locations. Combining regions reduces the total entropy, as expected when distinguishable particles become indistinguishable, and as required by a theorem in quantum mechanics for sub-additive entropy. Combining large volumes of small regions gives the usual entropy of mixing for a semi-classical ideal gas, resolving Gibbs paradox without invoking quantum symmetry for particles that may be meters apart. Other models presented here are based on Ising-like spins, which are solved analytically in one dimension. Focusing on the bonds between the spins, we find 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 fully-accurate predictions. Another result using Ising-like spins involves simulations that include a nonlinear correction to Boltzmann’s factor, which mimics the statistics of indistinguishable states by imitating the dynamics of spin exchange on intermediate lengths. These simulations exhibit 1/f-like noise at low frequencies (f), and white noise at higher f, similar to the equilibrium thermal fluctuations found in many materials.

Shift a laser beam back and forth to exchange heat and work in thermodynamics

Although the equivalence of heat and work has been unveiled since Joule’s ingenious experiment in 1845, they rarely originate from the same source in experiments. In this study, we theoretically and experimentally demonstrated how to use a high-precision optical feedback trap to combine the generation of virtual temperature and potential to simultaneously manipulate the heat and work of a small system. This idea was applied to a microscopic Stirling engine consisting of a Brownian particle under a time-varying confining potential and temperature. The experimental results justified the position and the velocity equipartition theorem, confirmed several theoretically predicted energetics, and revealed the engine efficiency as well as its trade-off relation with the output power. The small theory–experiment discrepancy and high flexibility of the swift change of the particle condition highlight the advantage of this optical technique and prove it to be an efficient way for exploring heat and work-related issues in the modern thermodynamics for small systems.

Chemical Potential Differences in the Macroscopic Limit from Fluctuations in Small Systems.

We present a new method for computing chemical potential differences of macroscopic systems by sampling fluctuations in small systems. The small system method, presented by Schnell et al. [Schnell et al., J. Phys. Chem. B, 2011, 115, 10911], is used to create small embedded systems from molecular dynamics simulations, in which fluctuations of the number of particles are sampled. The sampled fluctuations represent the Boltzmann distributed probability of the number of particles. The overlapping region of two such distributions, sampled from two different systems, is used to compute their chemical potential difference. Since the thermodynamics of small systems is known to deviate from the classical thermodynamic description, the particle distributions will deviate from the macroscopic behavior as well. We show how this can be utilized to calculate the size dependence of chemical potential differences and eventually extract the chemical potential difference in the thermodynamic limit. The macroscopic chemical potential difference is determined with a relative error of 3% in systems containing particles that interact through the truncated and shifted Lennard-Jones potential. In addition to computing chemical potential differences in the macroscopic limit directly from molecular dynamics simulation, the new method provides insights into the size dependency that is introduced to intensive properties in small systems.

Adsorption of an Ideal Gas on a Small Spherical Adsorbent.

The ideal gas model is an important and useful model in classical thermodynamics. This remains so for small systems. Molecules in a gas can be adsorbed on the surface of a sphere. Both the free gas molecules and the adsorbed molecules may be modeled as ideal for low densities. The adsorption energy, , plays an important role in the analysis. For small adsorbents this energy depends on the curvature of the adsorbent. We model the adsorbent as a sphere with surface area , where R is the radius of the sphere. We calculate the partition function for a grand canonical ensemble of two-dimensional adsorbed phases. When connected with the nanothermodynamic framework this gives us the relevant thermodynamic variables for the adsorbed phase controlled by the temperature T, surface area , and chemical potential . The dependence of intensive variables on size may then be systematically investigated starting from the simplest model, namely the ideal adsorbed phase. This dependence is a characteristic feature of small systems which is naturally expressed by the subdivision potential of nanothermodynamics. For surface problems, the nanothermodynamic approach is different, but equivalent to Gibbs’ surface thermodynamics. It is however a general approach to the thermodynamics of small systems, and may therefore be applied to systems that do not have well defined surfaces. It is therefore desirable and useful to improve our basic understanding of nanothermodynamics.