Where does mixing collide?

The entropy of mixing is the rise in entropy that occurs when two or more distinct substances, at first kept apart, are allowed to intermingle, a change that is irreversible for substances that can be told apart. Gibbs's 1876 On the Equilibrium of Heterogeneous Substances derived this quantity from the composition of a mixture and showed that combining two different gases raises entropy by an amount that does not depend on how alike the gases are, while combining a gas with more of itself raises it by nothing, the puzzle now called the Gibbs paradox. The referent here is a thermodynamic state function scaled in energy per unit temperature, and it has nothing to do with the loss of memory in a stirred dynamical system or the pattern of contacts between social groups.[1]

Unit

A change in entropy on combining substances, in joules per kelvin (or per mole)

Recovery means

Not applicable: for substances that can be told apart the process is thermodynamically irreversible

Measured by

Computed from the mole fractions of the components, for ideal mixing as minus R times the sum of each fraction times its logarithm

In the study of dynamical systems, mixing is a strong form of randomization in which a system loses track of its starting configuration as time passes, so that any region of the state space becomes, over the long run, spread evenly through every other region. Gibbs's 1902 Elementary Principles in Statistical Mechanics gave the intuitive picture with its image of a drop of dark ink stirred into water until the fluid looks uniformly grey, and the property was later given a precise measure-theoretic definition, set out for example in Walters's 1982 An Introduction to Ergodic Theory, as the statistical independence of a set from its own image after long times. The referent is a mathematical property of a transformation on a probability space, and it is distinct from the entropy released when substances intermingle or the contact patterns that carry a disease.[2][3]

Unit

A property a measure-preserving transformation either has or lacks, defined as a limit over long times

Recovery means

Not applicable

Measured by

Established by proof, or estimated numerically through the decay of correlations between a function and its time-shifted values

Social mixing is the pattern of who comes into contact with whom across a population, the structure of interaction between age groups or other categories that governs how far and how fast a transmissible infection spreads. Nold's 1980 Heterogeneity in disease-transmission modeling introduced proportionate mixing to represent these unequal contact rates inside transmission models, and Mossong and colleagues' 2008 Social contacts and mixing patterns relevant to the spread of infectious diseases measured them directly by surveying tens of thousands of daily contacts across eight countries. The referent is an empirical pattern of human contact, counted as contacts per person per day between groups, and it is unrelated to the entropy of combining substances or the randomization of a dynamical system.[4][5]

Unit

A rate of contacts per person per day between groups, often arranged as a contact matrix

Recovery means

Not applicable

Measured by

Estimated from contact diaries and population surveys, then encoded as a mixing matrix used in transmission models