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Register a new userThe Big World of Nanothermodynamics
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Authors
Ralph V. Chamberlin
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Nanothermodynamics extends standard thermodynamics to facilitate finite-size effects on the scale of nanometers. A key ingredient is Hills 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 Boltzmanns 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.
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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.
The Pair Approximation method has been formulated for the isotropic ferromagnetic Heisenberg model with spin $S=1$. The exchange interactions of arbitrary range have been taken into account. The single-ion anisotropy has been considered as well as the external magnetic field. Within the method, the Gibbs free-energy has been derived, from which all thermodynamic properties can be self-consistently obtained. In order to illustrate the developed formalism, the numerical calculations have been performed for CrIAs planar magnetic semiconductor, a hypothetical material whose existence has been recently predicted by the Density Functional Theory-based calculations. For this model material, all the relevant thermodynamic magnetic properties have been studied. The numerical results have been presented in the figures and discussed.
A parallel implementation of coupled spin-lattice dynamics in the LAMMPS molecular dynamics package is presented. The equations of motion for both spin only and coupled spin-lattice dynamics are first reviewed, including a detailed account of how magneto-mechanical potentials can be used to perform a proper coupling between spin and lattice degrees of freedom. A symplectic numerical integration algorithm is then presented which combines the Suzuki-Trotter decomposition for non-commuting variables and conserves the geometric properties of the equations of motion. The numerical accuracy of the serial implementation was assessed by verifying that it conserves the total energy and the norm of the total magnetization up to second order in the timestep size. Finally, a very general parallel algorithm is proposed that allows large spin-lattice systems to be efficiently simulated on large numbers of processors without degrading its mathematical accuracy. Its correctness as well as scaling efficiency were tested for realistic coupled spin-lattice systems, confirming that the new parallel algorithm is both accurate and efficient.
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 Boltzmanns 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.
The structural arrest of a polymeric suspension might be driven by an increase of the cross--linker concentration, that drives the gel transition, as well as by an increase of the polymer density, that induces a glass transition. These dynamical continuous (gel) and discontinuous (glass) transitions might interfere, since the glass transition might occur within the gel phase, and the gel transition might be induced in a polymer suspension with glassy features. Here we study the interplay of these transitions by investigating via event--driven molecular dynamics simulation the relaxation dynamics of a polymeric suspension as a function of the cross--linker concentration and the monomer volume fraction. We show that the slow dynamics within the gel phase is characterized by a long sub-diffusive regime, which is due both to the crowding as well as to the presence of a percolating cluster. In this regime, the transition of structural arrest is found to occur either along the gel or along the glass line, depending on the length scale at which the dynamics is probed. Where the two line meet there is no apparent sign of higher order dynamical singularity. Logarithmic behavior typical of $A_{3}$ singularity appear inside the gel phase along the glass transition line. These findings seem to be related to the results of the mode coupling theory for the $F_{13}$ schematic model.
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