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The zeroth law of thermodynamics states that if two thermodynamic systems are each in thermal equilibrium with a third, then all three are in thermal equilibrium with each other. Two systems are said to be in the relation of thermal equilibrium if they are linked by a wall permeable only to heat, and do not change over time.[1] As a convenience of language, systems are sometimes also said to be in a relation of thermal equilibrium if they are not linked so as to be able to transfer heat to each other, but would not do so if they were connected by a wall permeable only to heat. The law is important for the mathematical formulation of thermodynamics, which needs the assertion that the relation of thermal equilibrium is an equivalence relation. A system is said to be in thermal equilibrium when it experiences no net change of its observable state over time. This statement asserts that thermal equilibrium is a Euclidean relation between thermodynamic systems. If two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C. In the space of thermodynamic parameters, zones of constant temperature form a surface, that provides a natural order of nearby surfaces. In Caratheodory's (1909) theory, it is postulated that there exist walls "permeable only to heat", though heat is not explicitly defined in that paper. These ideas may be regarded as helping to clarify the physical meaning of the usual statement of the zeroth law of thermodynamics.
They then proposed that "it may be shown to follow that the condition for thermal equilibrium between several assemblies is the equality of a certain single-valued function of the thermodynamic states of the assemblies, which may be called the temperature t, any one of the assemblies being used as a "thermometer" reading the temperature t on a suitable scale. An effective temperature ?, conjugated to a generalized entropy s(q), was introduced recently for a system of interacting particles. Zeroth Law of Thermodynamics The "zeroth law" states that if two systems are at the same time in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
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The most precise statement of the zeroth law is that thermal equilibrium constitutes an equivalence relation on pairs of thermodynamic systems. If we also grant that all thermodynamic systems are in thermal equilibrium with themselves, then thermal equilibrium is also a reflexive relation.

An equivalence relationship on a set (such as the set of thermally equilibrated systems) divides that set into a collection of distinct subsets ("disjoint subsets") where any member of the set is a member of one and only one such subset. One may therefore construct a global temperature function that provides a continuous ordering of states. Nevertheless, this usual statement perhaps does not explicitly convey the full physical meaning that underlies it. Another modern expression of this idea is that "All diathermal walls are equivalent".[11] This might also be expressed by saying that there is precisely one kind of non-mechanical, non-matter-transferring contact equilibrium between thermodynamic systems. It is the opinion of Lieb and Yngvason (1999) that the derivation from statistical mechanics of the law of entropy increase is a goal that has so far eluded the deepest thinkers.[12] Thus the idea remains open to consideration that the existence of heat and temperature are needed as coherent primitive concepts for thermodynamics, as expressed, for example, by Maxwell and Planck.
Fowler invented the title 'the zeroth law of thermodynamics' when he was discussing the 1935 text of Saha and Srivastava. This postulate of the "Existence of temperature" could with advantage be known as the zeroth law of thermodynamics". It applies to systems composed of many particles, however, we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath. Using driven stochastic lattice gases, we ask whether consistent definitions of an effective chemical potential ?, and an effective temperature Te, are possible. The screenshot came from the Death Note anime series, and the blood was provided by KeRen-R. In other words, the set of all equilibrated thermodynamic systems may be divided into subsets in which every system belongs to one and only one subset, and is in thermal equilibrium with every other member of that subset, and is not in thermal equilibrium with a member of any other subset. In the case of the zeroth law, these subsets consist of systems which are in mutual equilibrium. The dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters, thus, for an ideal gas described with three thermodynamic parameters P, V and n, it is a two-dimensional surface. The first sentence of this present article is a version of this statement.[16] It is not explicitly evident in the existence statement of Fowler and Guggenheim that temperature refers to a unique attribute of a state of a system, such as is expressed in the idea of the hotness manifold.
A Survey of Thermodynamics, American Institute of Physics Press, New York, ISBN 978-0-88318-797-5. The physics and mathematics of the second law of thermodynamics, Physics Reports, 310: 1–96.
Chapter 1, 'An Outline of Thermodynamical Structure', pages 3–32, in New Perspectives in Thermodynamics, edited by J.
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This means that a unique "tag" can be assigned to every system, and if the "tags" of two systems are the same, they are in thermal equilibrium with each other, and if they are not, they are not.
One consequence of this reasoning is that thermal equilibrium is a transitive relationship: If A is in thermal equilibrium with B and B is in thermal equilibrium with C, then A is in thermal equilibrium with C.
This partitioning allows any member of the subset to be uniquely "tagged" with a label identifying the subset to which it belongs. They presume that temperature is a physical quantity and then deduce the statement "If a body A is in temperature equilibrium with two bodies B and C, then B and C themselves will be in temperature equilibrium with each other".
Also their statement refers explicitly to statistical mechanical assemblies, not explicitly to macroscopic thermodynamically defined systems. A version of Statistical Mechanics for Students of Physics and Chemistry, first printing 1939, reprinted with corrections 1965, Cambridge University Press, Cambridge UK.
Ultimately, this property is used to justify the use of thermodynamic temperature as a tagging system. Another consequence is that the equilibrium relationship is symmetric: If A is in thermal equilibrium with B, then B is in thermal equilibrium with A.
Although the labeling may be quite arbitrary,[8] temperature is just such a labeling process which uses the real number system for tagging.
This is signaled in the postulate of this paper of Caratheodory that precisely one non-deformation variable is needed to complete the specification of a thermodynamic state, beyond the necessary deformation variables, which are not restricted in number. Thermodynamic temperature provides further properties of thermally equilibrated systems, such as order and continuity with regard to "hotness" or "coldness", but these properties are not implied by the standard statement of the zeroth law. Thus we may say that two systems are in thermal equilibrium with each other, or that they are in mutual equilibrium. The zeroth law justifies the use of suitable thermodynamic systems as thermometers to provide such a labeling, which yield any number of possible empirical temperature scales, and justifies the use of the second law of thermodynamics to provide an absolute, or thermodynamic temperature scale.

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