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The two-dimensional systems of vortices confined to a finite area can form thermal equilibrium states at negative temperature, [17] [18] and indeed negative temperature states were first predicted by Onsager in his analysis of classical point vortices. [19]
Heat equation. Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. The height and redness indicate the temperature at each point. The initial state has a uniformly hot hoof-shaped region (red) surrounded by uniformly cold region (yellow). As time passes the heat diffuses into the cold region.
Point vortices confined to finite area were predicted by Onsager to exhibit states of negative temperature. [11] [12] This possibility of negative absolute temperature can be traced to the finite phase space of the point vortex system: in contrast to a massive particle moving on a plane, each point vortex only has two degrees of freedom.
The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases ...
A number of materials contract on heating within certain temperature ranges; this is usually called negative thermal expansion, rather than "thermal contraction".For example, the coefficient of thermal expansion of water drops to zero as it is cooled to 3.983 °C (39.169 °F) and then becomes negative below this temperature; this means that water has a maximum density at this temperature, and ...
Assuming that the concentration of fermions does not change with temperature, then the total chemical potential μ (Fermi level) of the three-dimensional ideal Fermi gas is related to the zero temperature Fermi energy E F by a Sommerfeld expansion (assuming ): = + [() +], where T is the temperature.
2-dimensional Hubbard model. The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. [1] It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each electron experiences competing forces: one pushes it to tunnel to ...
On the empirical temperature scales that are not referenced to absolute zero, a negative temperature is one below the zero-point of the scale used. For example, dry ice has a sublimation temperature of −78.5 °C which is equivalent to −109.3 °F. [97] On the absolute Kelvin scale this temperature is 194.6 K.