Total entropy production and its three constituent components are described both as fluctuating trajectory-dependent quantities and as averaged contributions in the context of the continuous Markovian dynamics, described by stochastic differential equations with multiplicative noise, of systems with both odd and even coordinates with respect to time reversal, such as dynamics in full phase space. Two of these constituent quantities obey integral fluctuation theorems and are thus rigorously positive in the mean due to Jensen's inequality. The third, however, is not and furthermore cannot be uniquely associated with irreversibility arising from relaxation, nor with the breakage of detailed balance brought about by nonequilibrium constraints. The properties of the various contributions to total entropy production are explored through the consideration of two examples: steady-state heat conduction due to a temperature gradient, and transitions between stationary states of drift diffusion on a ring, both in the context of the full phase space dynamics of a single Brownian particle.

The nucleation theorems relate the temperature and supersaturation dependence of the rate of nucleation of droplets from a metastable vapor phase to properties of the critical molecular cluster, the size that is approximately equally likely to grow or decay. They are derived here using a combination of statistical mechanics and cluster population dynamics, using an arbitrary model cluster definition. The theorems are employed to test the validity of the classical theory of homogeneous nucleation and its “internally consistent” form. It is found that the properties of the critical cluster for these models are incorrect, and it emerges that this occurs because the classical theory employs the free energy of a fixed droplet, rather than one free to take any position in space. Thus a term representing positional, or mixing, entropy is missing from the cluster free energy. A revised model is proposed, based on the capillarity approximation but with such a term included, and it is shown that it is fully consistent with the nucleation theorems. The model increases classical rates by factors of approximately ${10}^4$–${10}^6$. Other nucleation models should be tested for internal consistency using the same methods. Finally, the nucleation theorems are used to extract the excess internal energies of molecular clusters from experimental data for several substances.

A conceptual framework to predict microphysical and optical properties of contrail particles within a wingspan behind the source aircraft is developed. Results from two decades of contrail observations and numerical simulations are reviewed forming the basis of theoretical model development. The model utilizes cloud theory applied to the dynamics and thermodynamics of jet aircraft exhaust plumes in upper tropospheric conditions. Droplet nuclei include soot particles emitted from aircraft engines and atmospheric particles entrained into the plume. These precursor particles activate into copious homogeneously freezing water droplets as the plume relative humidity rises beyond liquid water saturation. A unimodal size spectrum of ice particles develops wherein ice particles grow to micrometer mean sizes. Contrail particle formation is analyzed over a wide range of soot emissions relating to conventional jet fuels as well as to alternative aviation fuels producing much less soot and volatile particle emissions. For current aviation fuels and propulsion technology, the number of contrail ice particles scales roughly in proportion to the number of emitted soot particles that act as water condensation nuclei despite their poor hygroscopicity. Close to the contrail formation threshold, only few plume particles can be water activated and freeze. Implications for effects of alternative fuels on contrails, an arena for future scientific exploration, are outlined.

The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, while the second two comprise the housekeeping heat. We denote these two components the transient and generalized housekeeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal, which restricts consideration of directional variables such as velocity.

The principles of statistical mechanics have been used to develop a theory of the nucleation of a phase transition, but a number of subtle questions remain and are highlighted in this review. A central issue is the cluster definition, the mathematical scheme which distinguishes a molecular cluster from a collection of separate molecules. There is also the question of whether thermodynamic transition state theory is suitable to describe the process, or whether the development of kinetic rate equations describing cluster growth and decay is a better approach. The classical theory of nucleation is flawed but appears to provide useful estimates in some cases, including water. Phenomenological extensions of the classical theory can improve matters. However, improvements in the theory from microscopic considerations are not simple to apply, require major computational effort and suffer from uncertainties due to lack of knowledge of the fundamental intermolecular interactions. Calculations of the nucleation rate of water droplets are especially difficult since this substance is notoriously difficult to model. Nevertheless, as capabilities improve, accurate calculations should come within reach, which will offer better understanding of the process for practical applications such as the transition from dry to wet steam.