Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in thermal equilibrium with itself if the temperature within the system is spatially uniform and temporally constant.
The relation of thermal equilibrium is an instance of equilibrium between two bodies, which means that it refers to transfer through a selectively permeable partition of matter or work; it is called a diathermal connection. According to Lieb and Yngvason, the essential meaning of the relation of thermal equilibrium includes that it is reflexive and symmetric. It is not included in the essential meaning whether it is or is not transitive. After discussing the semantics of the definition, they postulate a substantial physical axiom, that they call the "zeroth law of thermodynamics", that thermal equilibrium is a transitive relation. They comment that the equivalence classes of systems so established are called isotherms.[1]
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One may imagine an isolated system, initially not in its own state of internal thermal equilibrium. It could be subjected to a fictive thermodynamic operation of partition into two subsystems separated by nothing, no wall. One could then consider the possibility of transfers of energy as heat between the two subsystems. A long time after the fictive partition operation, the two subsystems will reach a practically stationary state, and so be in the relation of thermal equilibrium with each other. Such an adventure could be conducted in indefinitely many ways, with different fictive partitions. All of them will result in subsystems that could be shown to be in thermal equilibrium with each other, testing subsystems from different partitions. For this reason, an isolated system, initially not its own state of internal thermal equilibrium, but left for a long time, practically always will reach a final state which may be regarded as one of internal thermal equilibrium. Such a final state is one of spatial uniformity or homogeneity of temperature.[3] The existence of such states is a basic postulate of classical thermodynamics.[4][5] This postulate is sometimes, but not often, called the minus first law of thermodynamics.[6] A notable exception exists for isolated quantum systems which are many-body localized and which never reach internal thermal equilibrium.
If an initially isolated physical system, without internal walls that establish adiabatically isolated subsystems, is left long enough, it will usually reach a state of thermal equilibrium in itself, in which its temperature will be uniform throughout, but not necessarily a state of thermodynamic equilibrium, if there is some structural barrier that can prevent some possible processes in the system from reaching equilibrium; glass is an example. Classical thermodynamics in general considers idealized systems that have reached internal equilibrium, and idealized transfers of matter and energy between them.
Another explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by B. C. Eu. He considers two systems in thermal contact, one a thermometer, the other a system in which several irreversible processes are occurring. He considers the case in which, over the time scale of interest, it happens that both the thermometer reading and the irreversible processes are steady. Then there is thermal equilibrium without thermodynamic equilibrium. Eu proposes consequently that the zeroth law of thermodynamics can be considered to apply even when thermodynamic equilibrium is not present; also he proposes that if changes are occurring so fast that a steady temperature cannot be defined, then "it is no longer possible to describe the process by means of a thermodynamic formalism. In other words, thermodynamics has no meaning for such a process."[24] 2ff7e9595c
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