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As far as I remember, many textbooks on statistical physics introduce temperature as a condition of equilibrium of a composite thermodynamic system. E.g., if the system consists of two parts with energies $E_1,E_2$, and the total energy is fixed, $E=E_1+E_2$, then maximizing the entropy (and neglecting the small contributions due to the interactions on the systems borders) we have: $$ \frac{d}{dE_1}\left[S_1(E_1) + S_1(E-E_1)\right]=S_1'(E_1) - S_2'(E-E_1)=0\\\Rightarrow S_1'(E_1) = S_2'(E-E_1)\\\Rightarrow E_1^*, \frac{1}{T}=S_1'(E_1^*)=S_2'(E-E_1^*) $$ The temperature is thus a value of the derivative of the entropy (logarithm of the number of the microstates at thermal equilibrium, but not the derivative itself (the difference between a function and a value of this function at a specific point, contrary to what is suggested in this question, which motivated me to ask mine.)

What I miss here is the transition from this definition of temperature to the condition of thermal equilibrium as the equality of temperatures - this means that the temperature must be defined for each subsystem independently. So, is the temperature a function (of energy) or a value of a function at thermal equilibrium?

Roger V.
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  • I don't understand. If you see $T$ as a function, i.e. $T: E\mapsto T(E)$, then it maps an energy to the (equilibrium) temperature of a system. Then $T(E)$ is the temperature of the system, while $T$ is a function which maps an energy to the associated temperature. At least in equilibrium thermodynamics, all quantities anyway are defined for equilibrium only. Maybe you can clarify to me what I am missing. – Tobias Fünke Jun 08 '22 at 16:59
  • The condition on equilibrium could then be stated like: The temperature function of system 1, $T_1$, evaluated at $E_1$ must be equal to the temperature function of system 2. $T_2$ evaluated at $E_2=E-E_1$, i.e. $T_1 (E_1 ) = T_2(E_2)$. – Tobias Fünke Jun 08 '22 at 17:04
  • @JasonFunderberker if it is a function, the temperature can be defined even for an isolated system (mucrocanonical ensemble). And it is not specifically related to equilibrium - it is essentially the density of states – Roger V. Jun 08 '22 at 17:04
  • But if you have the function, you have all temperature values for the corresponding energies and vice versa: Define $1/T:=S^\prime$ and the equilibrium condition you derive follows. If you start from your definition (i.e. start from the value of the derivative of $S$), then you can do it for a different range of energies, thereby defining a function by setting $1/T(E) = S^\prime (E)$. BTW: In principle you should have two temperature functions, no? Because $S_1 \neq S_2$ as functions, in general and the same should hold for their corresponding derivatives. – Tobias Fünke Jun 08 '22 at 17:10
  • @JasonFunderberker the question is about the rigorous definition of the concept. Is temperature a value or a function? Or do we have two things called temperature? Can we define temperature for a closed system? Is it a state variable? Perhaps the question is silly or perhaps it is some fine point skipped over in textbooks - but it seems like things are ambiguous here. – Roger V. Jun 08 '22 at 18:18
  • Maybe I'm misunderstanding, your question but temperature is a value that represents the average kinetic energy of a system. The condition assumed for a stable temperature value, is that the average kinetic energy at measurement is a constant. But a constant average kinetic energy is an equilibrium condition. Is this what you are after? – Stevan V. Saban Jun 08 '22 at 18:36
  • Okay, but what is the difference to the following question: Is kinetic energy (of a free particle in 1D) a function or a value? $E := v^2/2m$. Or is force a function or a value? Because for a potential $V$ we have $F(x)= -V^\prime (x)$. Could you explain me the differences to your question? I really still don't understand yet. – Tobias Fünke Jun 08 '22 at 18:42
  • @SteveSaban I think my definition is more general. You are talking about equipartition theorem, and only for the case of non-interacting particles - since otherwise the energy is not merely kinetic. However, if the temperature is energy (per particle), it is not a function of energy, hut a value. – Roger V. Jun 08 '22 at 18:43
  • @JasonFunderberker we're taling specifically about it being a function of energy. I am looking for a rigorous/standard definition of temperature. Is it a value that characterizes thermal equilibrium or is it a function of the density-of-states whose value is equal for all the systems in thermal equilibrium or is it average energy per degree of freedom. I know the math, I know to solve problems - I am just surprized that such a widely used concept is not clearly defined. You seem to be unsure about the canonical definition either... – Roger V. Jun 08 '22 at 18:50
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    "I am just surprized that such a widely used concept is not clearly defined." The various definitions of entropy are worse. – Stevan V. Saban Jun 08 '22 at 19:31
  • @SteveSaban indeed – Roger V. Jun 08 '22 at 19:39
  • It is both. Just like pressure $P$ or kinetic energy $K$, temperature is both a function and can have a specific value. Temperature is generally defined as a function of the energy, and a system at equilibrium will have a specific value for that function. If two interacting systems have the same value, they are in thermal equilibrium. – ratsalad Jun 07 '23 at 14:03
  • @ratsalad I think you miss the nuance: it is not about a function (as a dependence) vs. value of this function, it is about how we define temperature in stat physics. If we define it as a point characterizing equilibrium between two systems, it cannot be a function of system parameters (in other words, is a root of an equation a function?) – Roger V. Jun 07 '23 at 14:28
  • I still think the answer is simple: temperature is defined fundamentally in thermo as a function. Thermal equilibrium between two interacting systems is found to be when the respective values of the temperature function for each of the systems are equivalent. Maybe you are conflating the definition of temperature with thermal equilibrium? – ratsalad Jun 07 '23 at 14:39
  • @ratsalad I think there are different definitions of temperature, depending on what one postulates (the text one reads) and whether we talk about stat.phys or thermo - just as with entropy. – Roger V. Jun 07 '23 at 15:19
  • @ratsalad if temperature is a number characterizing equilibrium between two bodies, what is the temperature of an isolated object then? Does it have one? What if the object is not isolated, but out of equilibrium with surroundings, like Sun - does it have a temperature? – Roger V. Jun 07 '23 at 15:25
  • Yes, an isolated object has a temperature. A common scenario in classical and stat thermo is considering two isolated systems each with its own temp. If both have the same temp, then if you bring them into contact and allow them to interact, you know they will already be at thermal equilibrium. – ratsalad Jun 07 '23 at 16:08
  • Out-of-equilibrium systems do not have a formally defined temperature in equilibrium thermodynamics. But often they can be considered to be approximately in equilibrium. In fact, nothing in reality can ever be at true equilibrium (or isolated). Thermo is all about ideal, limiting cases. – ratsalad Jun 07 '23 at 16:11
  • Also, when being careful, it is best not to think that "temperature is a number characterizing equilibrium between two bodies", even though we often casually talk that way. Temperature is a function, and if two bodies are in thermal equilibrium, the values of their temperature functions will be equal. – ratsalad Jun 07 '23 at 16:14
  • @RogerVadim Maybe this helps -- Temperature is the unique state function with the following property: if the values of this function for different interacting systems are equal, then the systems are necessarily in thermal equilibrium. (And by "unique" here I mean the class of bijective functions with this property). The stat mech example in your OP just uses this property to uniquely identify the temperature function. – ratsalad Jun 07 '23 at 17:18

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The definition of temperature is $$ T = \left(\frac{\partial S}{\partial E}\right)_{V,N} $$ and as such it is a function of $E$, $V$ and $N$: $$ T = T(E,V,N) $$ The value of $T$ at a particular state is the numerical value of this function at the energy, volume and number of moles in that state.

If you have two systems you can can calculate their temperatures. If they are equal, the systems are in thermal equilibrium.

In other words, the equilibrium condition requires the value of tyemperature to be the same in both systems, not the functions: $$ \underbrace{T(E_1,V_1,N_1)}_{T_1} = \underbrace{T(E_2,V_2,N_2)}_{T_2} $$ which we write more conventionally as $$T_1 = T_2$$

I suspect the confusion arises from the common mathematical notation, $v = v(t)$, which assigns the same symbol $v$ to both the function (e.g., velocity as a function of time) and the value of the function at some specific $t$ (as in $v=0$ at $t=0$).

Themis
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