By Andy May

I never thought I would participate in a serious debate on the definition of “temperature.” But it happened on twitter, and with people who have degrees in physics and other hard sciences! Since I worked with kinetic and effective temperature estimates for 42 years as a petrophysicist, I knew these topics intimately and never gave them a second thought. So, to hear people say (paraphrasing) that they were not really temperatures came as a shock. Their proposed sole definition of “temperature:” is the thermodynamic statistical definition:

Equation 1:

Where T is temperature, U is the internal energy, S is entropy, ∂U/∂S is the partial derivative of internal energy with respect to entropy, N is the number of particles, and V is the volume. The derivative is taken while keeping the volume and number of particles fixed, thus it is an equilibrium temperature and has no meaning outside the laboratory. Besides, entropy is a well-defined state function only in equilibrium thermodynamics; outside equilibrium, entropy can be defined statistically (see Boltzmann), but it is not unique and does not serve the same role.

Equation 1 can be called a definition of the “equilibrium temperature,” and when a system is in equilibrium it is an exact definition. It follows directly from the first law of thermodynamics. The first law states that energy cannot be created or destroyed, only converted from one form to another. That said, the first law is not the sole definition of temperature, as the word is used today. There are many other temperature definitions in common usage as well as in science. Let’s look at some of them:

Kinetic temperature

You can measure the instantaneous temperature of molecules in a shock wave, a plasma, a turbulent flow, or a laser-heated gas. It is a function of the mean kinetic energy of the molecules in the gas (or other fluid) (Reif, 2009). This temperature is widely used in shock physics, plasma physics, petrophysics (my former profession), and in molecular dynamics simulations. It is a valid temperature that does not require thermodynamic equilibrium.

Effective temperature

There are some systems that have no equilibrium temperature at all. Yet, we routinely define an effective temperature for these systems (Herzberg, 1950). These include turbulence, granular materials, active matter, moving dissipative systems, etc. These are legitimate temperatures and useful.

Non-equilibrium systems have a temperature, sometimes more than one depending upon the parameters used. For example, a molecular beam can have a vibrational or rotational temperature and they can be different. Plasmas may have an electron temperature or an ion temperature. NMRs have a spin temperature. Electronic circuits have a noise temperature. These temperatures are meaningful and important especially when the system is far from equilibrium.

Local temperature

This is the temperature used in climate and meteorological studies. It is the foundation of heat conduction, and hydrodynamics. One assumes local thermodynamic equilibrium over some volume and measures or assumes a local temperature for the volume. This is the logic used in the famous Navier-Stokes equations. This assumed “local equilibrium” is valid over small volumes (for example an “air parcel”) for short time periods. The problem with some climate models is that they assume “local equilibrium” for volumes and time periods that are too large and too long.

Discussion

Temperature is not a primitive mechanical property like mass; it is an emergent statistical property that characterizes the distribution of energy among degrees of freedom (Landau & Lifshitz, 1980). Mass can be measured independently of the rest of the universe or system and does not change as the system around it changes. Temperature follows from the 0th law of thermodynamics, which says:

If system A is in thermal equilibrium with system B,

and system B is in thermal equilibrium with system C,

then A is in thermal equilibrium with C.

The 0th law establishes that “temperature” is a meaningful physical quantity because thermal equilibrium is transitive. It does not define temperature, but it guarantees that temperature is meaningful and measurable. We often hear that temperature is transitive, but this is only true in laboratory settings or at equilibrium. Outside equilibrium, temperature can lose transitivity and different degrees of freedom can produce different temperatures. Transitivity is the basis of equilibrium temperature, but not all temperature measurements.

In summary, the thermodynamic definition of temperature applies only to equilibrium states, but physics uses many other temperature concepts like kinetic, effective, local, and generalized that are essential for describing real systems far from equilibrium. Restricting “temperature” to its equilibrium definition ignores the vast range of physical systems where temperature is well-defined and indispensable.

Some additional non-equilibrium temperature references are cited and discussed here.

Bibliography

Herzberg, G. (1950). Molecular Spectra and Molecular Structure. Volume I: Spectra of Diatomic Molecules. Second Edition. D. Van Nostrand.

Landau, L. D., & Lifshitz, E. M. (1980). Statistical Physics, Third Edition. Butterworth-Heinemann.

Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Waveland Press, Inc.