Thermodynamics is a system theory. And temperature is a result of the laws of thermodynamics. So you must accept the fact that this formulation will constantly deal with processes in systems.

Definitions of thermodynamic quantities are specified by the laws of thermodynamics. For example, the First Law defines the internal energy U of a system.

The Zeroth Law (so named because it was formulated after the First, Second, and Third Laws were in place) establishes the empirical temperature.

The Zeroth Law is

If systems A and B are both in equilibrium across a diathermal boundary with system C, then systems A and B will be in equilibrium with one another if separated by a diathermal wall.

A diathermal wall is a wall pervious to heat transfer. System C is the thermometer.

If there are no external fields present we may (essentially always) claim the system has two independent variables. Then the empirical temperature scale will be fixed by two constants. It is otherwise arbitrary. The choice is often conveniently (Celsius and Fahrenheit) of a linear scale.

The absolute thermodynamic tempreature is established from the Second Law. The original formulation of Carnot's Theorem (Clausius) identified the differential of the system property entropy S as equal to the heat transferred in a reversible process divided by the thermodynamic temperature T. To this is added the requirements of Clausius' Inequality that the change in entropy is positive definite.

The Thermodynamic temperature also satisfies the requirements of the empirical temperature in the Zeroth Law.

Iy you wish a more beautiful formulation of the thermodynamic temperature, I refer you to Caratheodory's theorem. I treat this in outline in my text [Helrich, 2008, Springer-Verlag]. There the inverse of the thermodynamic temperature is rigorously proven to be an integrating factor for the differential heat transferred in a system process.

The fact that absolute zero of thermodynamic temperature cannot be attained in a finite number of steps, i.e. is unattainable, is a consequence of the Third Law (Nernst's hypothesis). Stated more precisely it is a result of a corollary of the Third Law. The corollary is that absolute entropy approaches a constant for all substances in pure form as T approaches zero. In spite of what some authors claim, this is not a statement of the Third Law. It is a corollary.

So that is temperature for you. It is as rigorous as any other thermodynamic variable and a result of the laws of thermodynamics. But you really need all four laws if you want to do more than wave your hands.

Hope this helps.

It is measure of hotness or coldness of the body.

You are right that temperaturę could be given by relations ds.=dQ/T. However, temperature is related to kinetic energy of atoms or molecules. In this case, temperature is written by following equation: T=(2)/(f kB) where - mean kinetic energy, f - freedom degrees, kB - Boltzman constant.

Thank you for participating. This message isn't directed at either of you two anymore than it is to those who may think the question is trivial and perhaps choose not to participate for that reason. Temperature is not explained by equations for the reason given in my comments. Temperature was introduced as an arbitrary indefinable property and remains so today. The equations that include temperature are not telling us what temperature is, they are using the numerical value of an important thermodynamic property. That need is not in question. What is in question is a physical explanation of what is temperature? The temperature scale is a measure of the property of temperature. If it was a measure of some other property, then equations would need only to include that other property. I gave the example of Clausius' thermodynamic entropy to emphasize that it remains unexplained because temperature remains unexplained. Physicists should be able to explain both because they claim to explain both. The purpose of my question is to expose the circumstance that physicists have not explained temperature. Temperature was introduced as an indefinable property and remains an indefinable property today. An indefinable property is one that cannot be defined in terms of pre-existing properties. An indefinable unit is one that cannot be defined in terms of pre-existing units. 

temperature= a measurement of  the quantity energy in the system

Thermodynamics is a system theory. And temperature is a result of the laws of thermodynamics. So you must accept the fact that this formulation will constantly deal with processes in systems.

Definitions of thermodynamic quantities are specified by the laws of thermodynamics. For example, the First Law defines the internal energy U of a system.

The Zeroth Law (so named because it was formulated after the First, Second, and Third Laws were in place) establishes the empirical temperature.

The Zeroth Law is

If systems A and B are both in equilibrium across a diathermal boundary with system C, then systems A and B will be in equilibrium with one another if separated by a diathermal wall.

A diathermal wall is a wall pervious to heat transfer. System C is the thermometer.

If there are no external fields present we may (essentially always) claim the system has two independent variables. Then the empirical temperature scale will be fixed by two constants. It is otherwise arbitrary. The choice is often conveniently (Celsius and Fahrenheit) of a linear scale.

The absolute thermodynamic tempreature is established from the Second Law. The original formulation of Carnot's Theorem (Clausius) identified the differential of the system property entropy S as equal to the heat transferred in a reversible process divided by the thermodynamic temperature T. To this is added the requirements of Clausius' Inequality that the change in entropy is positive definite.

The Thermodynamic temperature also satisfies the requirements of the empirical temperature in the Zeroth Law.

Iy you wish a more beautiful formulation of the thermodynamic temperature, I refer you to Caratheodory's theorem. I treat this in outline in my text [Helrich, 2008, Springer-Verlag]. There the inverse of the thermodynamic temperature is rigorously proven to be an integrating factor for the differential heat transferred in a system process.

The fact that absolute zero of thermodynamic temperature cannot be attained in a finite number of steps, i.e. is unattainable, is a consequence of the Third Law (Nernst's hypothesis). Stated more precisely it is a result of a corollary of the Third Law. The corollary is that absolute entropy approaches a constant for all substances in pure form as T approaches zero. In spite of what some authors claim, this is not a statement of the Third Law. It is a corollary.

So that is temperature for you. It is as rigorous as any other thermodynamic variable and a result of the laws of thermodynamics. But you really need all four laws if you want to do more than wave your hands.

Hope this helps.

I agree with Helrich: thermodynamics, without its statistic-mechanical foundations, follows directly from empirical data, and this is its strength and also, formally, its weakness. In a more formal setting, I like particularly the definition of temperature in Caratheodory's axiomatic formulation of thermodynamics: 1/T is the integrating factor of heat, since dS=dQ/T, and dQ is an imperfect differential, while S is a state function.

The temperature of a system is the derivative of its energy with respect to its entropy, if the number of particles contained in the system and its volume remain fixed. This is the most fundamental and general definition of temperature. For systems comprised of non-interacting particles such as ideal gases, temperature can be expressed as T = 2/k, where E is the average kinetic energy per translational degree of freedom and k is Boltzmann's constant. But this latter definition is less fundamental and general. Probably the best explanations of temperature that I have ever seen are in Ralph Baierlein's two books "Atoms and Information Theory" and "Thermal Physics".

Please, take a look at papers:

Physical Review E 70. 055102(R) (2004)

and

Physica A 388(2009) 2122-2130

Thanks to Herr Stilck and Jack Demur. I tried to stay with the thermodynamic definition. The representation to which Demur refers is correct provided the brackets indicate the Gibbs ensemble average. In that case there is no real question about fundamentals, since ensemble theory is without approximation. If there is no interaction between the particles then, as Demur points out, T = 2/k. The issue is that the thermodynamic potentials internal energy U(S,V), Enthalpy H(S,P), Helmholtz Energy F(T,V) and Gibbs Energy G(T,P) result in thermodynamic temperature, pressure, entropy and volume defined as partial derivatives of these potentials. Essentially the only way to connect the microscopic picture to the thermodynamic picture is via the Helmholtz Energy, which is obtained as F = -kTln(Q(T,V)), where Q is the Partition function, which is the system phase space integral of exp(-H/kT) with H = Hamiltonian. There are here no approximations.The obvious snag is that T seems to have already appeared. Gibbs used the equality of temperature between two systems at equilibrium (essentially the future zeroth law) to show that a parameter he had called capital Theta was kT. There is a beautiful discussion in Gibbs' book Elementary Principles in Statistical Mechanics. But now we also have the statistical mechanical form (no approximations in ensemble theory) of a thermodynamic property as the ensemble average of the corresponding microscopic property, provided we can identify that. For U we can do this and U = = (kT)^2(dlnQ/d(kT)) at constant volume. Here we can see Demur's point that if H involves only momenta (no interactions) lnQ involves only an additive V and the simple form he cites results. He remains silent over the definition of temperature from the microscopic picture, since then we must begin to use cluster expansions and models of the interaction. We then introduce approximations and leave the realm of definitions. I tried to avoid all of this by staying with the thermodynamics. There is a connection, though.

There is no doubt abou it: temperature is a measuring quantity with respect to

a certain thermometer. Different thermometers show different temperatures  in

identical situations. Consquently, you need  a "generalized" temperature. The theoretical

way to it is described in Carl S. Helrich's contribution. But this absolute temperature

is only valid in equilibrium..Statistical physicists say: no problem, we define

1/T:=\partial S/partial U which is also valid in non-equilibrium by a statistical definition

of S. Correct, but are you sure that for this mathematical quantity a suitable

thermometer exists ? Clearly, the integrating factor in TdS="d"Q is only defined in

equilibrium. Why not define temperature in equilibrium and in non-equilibrium

phenomenologically ? The attached old paper may help. Another one is

 IntJEngngSci 15(1977)377-389. More recent papers are:

JNET 29(2004)237-255,    JNET 39(2014)113-121.

Also papers of David Jou are helpful.

Anyway, temperature is a macroscopic quantity which in special situations has a

statistical back-ground, but not in general, if the non-equilibrium entropy is not

specified. 

In the non-equilibrium context, let me add this reference Phys. Rev. E 93, 012121 (2016), where the authors discuss the concept of higher order temperature.

Temperature is a measurable property. That is not in question. The kelvin temperature scale is not in question. What is in question is that the degree Kelvin is an indefinable unit. The unit is indefinable and the property of temperature is indefinable. I have explained this. It is necessary for respondents to understand the difference between definable and indefinable. When a thermometer is inserted into a liquid which itself is in equilibrium, what is the physical event that occurs at the surface of the thermometer and which is named temperature? The original question is: What is temperature?  

undefined is not the same as undefinable, though. 

Dear Remi,

For a photon gas, T=q/3R(p)N, where q is the energy of the photon gas, N is the number of the photons, R(p)= [ζ(4)/ζ(3)]k,  whereζ(4) and ζ(3) are Riemann zeta functions, k is Boltzman's constant.

If E is considered as the average kinetic energy per degree of freedom, T=2E/k can only denote the temperature of an idea gas.

The temperature is a thermodynamic or statistical concept and has no analogue in classical mechanics. It is this concept that distinguishes thermodynamics from classical mechanics, and can be thought of as a conjugate to the entropy, just as the pressure can be thought of as conjugate to the volume. The relation dQ=TdS is only valid for a reversible process in thermodynamics and cannot be taken as a definition of T for irreversible processes. Despite this, the concept of temperature can be introduced during an irreversible process, but is not trivial. However, no matter how it is defined, its physical consequence must always be the following: Heat must always flow from hot to cold.

If you need more details, I can provide you later.

In thermodynamics, temperature is a state variable, all of the thermodynamic state variable should be able to explain from a micro perspective. A state variable should be defined according to its state property but not process.

Dear Tang Suye,

Entropy is a state variable defined in terms of process. Gibbs' formulation in terms of densities in Gamma space does not explain entropy. Boltzmann's formulation is based on kinetic theory of gases very close to equilibrium. It is not general.

Dear Tang Suye,

To be technically correct I should recognize that you are speaking of state variables not thermodynamic properties. Clausius' and later Gibbs' and Caratheodory's formulation of entropy relate dS to a reversible process. But there T only appears as an integrating factor with no fundamental definition in terms of the microscopic picture of matter, as has already been pointed out by others here. Any attempt to formulate T in terms of of the micro perspective, for which you call, involves an approximate formulation based ultimately on the Boltzmann approximation for the entropy of a gas. We may also use the Gibbs formulation, postulate the Hamiltonian as that for free point particles, and specify equilibrium. But the result is not then fundamental and cannot be used as a definition.

Gibbs classically warned us of the problem regarding our microscopic picture of matter in the introduction to he classic work on the Elementary Principles of Statistical Mechanics. We now know more than Gibbs did about the microscopic picture of matter.  But all of our attempts to formulate T in terms of a micro perspective will have approximations, including the quantum statistical results cited by our colleague above. None of these can be a definition if an approximation is involved.

Thermodynamics is a system science. In our minds' eyes we may believe that we know what a microscopic collection of atoms, molecules, or even Fermions or Bosons looks like. But we do not. Try to explain BEC in  pictorial terms. Our mathematics gives us only approximations, because we lack Pascal's requirement of super intelligence.

Dear Carl S. Helrich,

I have different opinions.  I think it is a deep issue, please see attached.  

Best regards

Is temperature in non-equilibrium defined ?

Eine sehr gute Frage. Temperature is based on the Zeroth Law dealing with equilibrium systems.  Traditionally it may be defined locally in spatial variation. But that's no answer.

James question is “what is temperature?”.  It should be answered by the definition of temperature, but we find it is not easy to obtain a definitive answer. Why？

By experience, temperature is an intensive variable which measure the intensity of thermal motion, by thermodynamics, we know that temperature is a macroscopic variable, all of the macroscopic variables should be able to explain from a micro perspective, otherwise, it will  not meet the requirements of a self-consistent theory.

The realistic situation is: many thermodynamic variables  can hardly be explained from a micro perspective, such as  temperature T,   enthalpy H, Helmholtz free energy F, grand potential …, etc,. why？

The real issue is: the energy classification for the internal energy U differs from that in  micro dynamics. Therefore  that,  thermodynamic variables can hardly be corresponded one to one with  micro dynamical variables.

The microscopic thermal motion within the system is a system property, temperature is an intensive variable which measure the intensity of thermal motion, but we don’t know which thermodynamic function can describe the energy of thermal motion.

It is a physics fact that temperature was accepted as one of four fundamentally indefinable properties. It, along with the other three, remains indefinable to this day. Presentations of evidence that it exists are not presentations of what it is. It exists without doubt while remaining indefinable. The usefulness of a magnitude of its measurement is clear. I gave the example of the dependency of Clausius' definition dS=(dQ/T). The practice of solving this and other such equations for temperature T do not yield definitions of temperature. The use in such equations of the indefinable temperature cannot be manipulated to undue temperature's indefinable status. Temperature occurs locally at the surface of the previously mentioned thermometer. Temperature is occurring locally everywhere in the previously mentioned liquid which is itself in equilibrium. Even if that liquid is not in equilibrium, temperature is occurring locally everywhere in it. What is it that is occurring that we call the property of temperature?  The coexistence of different scales for measuring temperature results from temperature's indefinable status. The units of an indefinable property are themselves indefinable units. They are not uniquely dictated by empirical evidence..They should be and could be. Yet they remain indefinable. As I pointed out early on, it is not known what it is that Clausius discovered when he established the existence of thermodynamic entropy.  The reason that his thermodynamic entropy cannot yet be explained is that physicists do not yet know what is temperature. The unexplained property of temperature cannot be explained by solving for temperature in terms of the unexplained property of thermodynamic entropy. My position is that temperature could have been and  should have been a defined property. However, progress in this direction is not possible so long as physicists no longer remember that they have not explained what temperature is. The other three recognized fundamentally indefinable properties also remain unexplained. The problem for theoretical physics is larger than learning what temperature is. The question presented though is: What is temperature? 

To James Putnam:

Please read the attachment and report.

Dear W. Muschik,

I have read it. What is the justification for arguing that there is a 'thermostatic' temperature for equilibrium conditions? There is a constant magnitude for temperature at all locations for equilibrium conditions, but, the use of the thermometer requires that temperature is dynamical. Empirical evidence consists of changes of velocities. Temperature is empirical evidence. I assume that this idea that there is a need for an additional concept of a 'dynamical' temperature results from the indefinable status of temperature. I say this because if temperature was made a defined property then its definition would have to be based upon the empirical evidence from which its existence is inferred. That empirical evidence is not static. The property of temperature is always dynamic and is not different for equilibrium conditions and non-equilibrium conditions. Measurements of its magnitude according to different temperature scales may be static for equilibrium conditions, but, the local conditions at the surface of the thermometer that give those readings are not static.  

It appears that the the question about the concept of temperature has elicited confusion as different commentators confine themselves to their own idea about what they mean by thermodynamics. It is certainly true that historically, the concept of temperature appeared in classical thermodynamics as a primitive concept, which does not require any definition. It is just like the concept of mass in mechanics. However, the scope of classical thermodynamics since the days of Maxwell, etc. has grown beyond what they had intended it to be. We now apply it to lattice systems, where particles are fixed in their positions so no motion is possible. In this case, we cannot use the average kinetic energy to quantify the temperature. Yet, these systems also possess a concept of temperature. Why do I say so? It is because, we use the zeroth law to introduce it as a number, which takes the same value in two systems in thermal equilibrium. A simple analysis shows that this number in the units of the Boltzmann constant can be identified with the inverse of the partial derivative of the entropy S with internal energy E. This defines the absolute temperature in equilibrium thermodynamics, which is valid even for a lattice model. In equilibrium, S is a state function of state variables such as E,V, etc. used in equilibrium. The scale of T determines the value of the entropy of the system or vice-versa.

Now, any standard textbook on thermodynamics will show that the absolute temperature introduced above can be related to daily usage temperatures in Celsius and Fahrenheit scales.  

There is a class of nonequilibrium states for which the entropy is also a state function, except that the state space now is enlarged by additional variables. This is nonequilibrium entropy now,. Again, a nonequilibrium temperature can be assigned and calculated by the inverse of the partial entropy-energy derivative in the enlarged space. This is what Dr. Muschik is suggesting as far as I understand. 

Other nonequilibrium states pose complications in defining their temperature, but I believe, it can be done. However, I will not talk about this complication here as it is beyond the scope of this thread.

I hope that my comments are useful to some of you.

I suggest that temperature is the same property without the need for separate treatment whether conditions are equilibrium or non-equilibrium. I further suggest that Clausius' thermodynamic entropy is the same property without the need for separate treatment whether conditions are equilibrium or non-equilibrium. These suggestions follow from my own theoretical work and from the continued condition that temperature is a fundamental indefinable property. The property is represented in physics equations only by its units. Its units of degrees are themselves indefinable units. It will be known when temperature is made a defined property because its units will, at that time, be made defined units. 

I have stated that Clausius' thermodynamic entropy remains unexplained. Its continued use in some responses causes me to invite those who use it to explain what it is. 

Dr. Putnam: I do not think there is much in dispute; only the interpretation is questionable. Clausius has no microscopic understanding of the entropy (presumably, this is what you mean by S being unexplained), but he had a very clear understanding of its thermodynamic significance. It was left to Boltzmann and Gibbs to provide its microscopic significance in terms of microstates of the system. One can debate whether Clausius entropy is the same or different than the microscopically defined entropy of Boltzmann and Gibbs. But this is not the issue here. We will stick with the Clausius entropy here. According to him, dS>dQ/T for an irreversible process, while the equality holds for a reversible process. The parameter T is the temperature of the heat bath, which is always assumed to be in equilibrium so that T is fixed, whether heat is transferred reversibly or not with the system of interest. The temperature T is a global property of the heat bath. In fact, nothing changes if the system is in equilibrium at some temperature T’ different from T or out of equilibrium. In the former case, one can establish that the heat flows from hot to cold. Here, T’ is a global property of the system. This is consistent with the fact that an equilibrium system is homogeneous so it has the same temperature throughout the system. All this is well known in classical thermodynamics and one can easily compute the irreversible entropy dS-dQ/T that is generated due to the heat flow. But in the latter case, it is not obvious how to define the temperature of the system, even though the change in the entropy S of the system is given by dS. Here, I agree with you that Clausius believes that S is the entropy of the system, whether it is in equilibrium or not. However, he can only compute dS for a reversible process so he can compute it only for a system in equilibrium. Because of this limitation, I also confess that many think that S exists only for systems in equilibrium. But Clausius gives no indication that T also exists as a global property for a system out of equilibrium. One problem is that a nonequilibrium system may be inhomogeneous so it cannot be described by a global parameter with the significance of a conventional interpretation of temperature in that case; sure, one can introduce locally varying temperatures for different regions, assuming that these regions are locally in equilibrium. This is what is traditionally done in local nonequilibrium thermodynamics. So you see that the temperature is hard to define for all nonequilibrium systems. However, it can be done with some care. Dr. Muschik has introduced the idea of a contact temperature, which is very close to what I have said.

My suggestion is that please go beyond Clausius; our understanding of nonequilibrium thermodynamics has grown since then, though it is not yet complete.

Going back to your concerns about units, I have to say that Nature provides us with scalars that are dimensionless. All units are man-made.     

Dear P. D. Gujrati,

"But Clausius gives no indication that T also exists as a global property for a system out of equilibrium. One problem is that a nonequilibrium system may be inhomogeneous so it cannot be described by a global parameter with the significance of a conventional interpretation of temperature in that case; sure, one can introduce locally varying temperatures for different regions, assuming that these regions are locally in equilibrium. This is what is traditionally done in local nonequilibrium thermodynamics. So you see that the temperature is hard to define for all nonequilibrium systems. However, it can be done with some care. Dr. Muschik has introduced the idea of a contact temperature, which is very close to what I have said."

The reading on a thermometer is not temperature. Temperature is happening at the submerged surface of the thermometer. Temperature is happening to the liquid inside the thermometer. The reading on the thermometer is a magnitude indicative of the amount of expansion that has resulted from temperature. The reading can be static, but temperature is not static. Temperature is not defined by that reading. A constant magnitude of a thermometer's reading in a liquid that is in equilibrium does not define temperature in that liquid. Temperature was introduced as a fundamentally indefinable property because it was understood since early physics that the reading on the thermometer is not what is temperature.  

"My suggestion is that please go beyond Clausius; our understanding of nonequilibrium thermodynamics has grown since then, though it is not yet complete."

There is no going beyond Clausius until it is learned what is was that Clausius discovered when he introduced thermodynamic entropy. The relevance of this lack of knowledge is that it continues because temperature remains unexplained. What is temperature?

"Going back to your concerns about units, I have to say that Nature provides us with scalars that are dimensionless. All units are man-made."  

If your assertion is that "All units are man-made." refutes my separation of units into definable units and indefinable units then you are mistaken. The difference between definable units and indefinable units is of monumental importance to theoretical  physics. In the case of temperature, its units of degrees are an admission that physicists were unable to learn what it was that empirical evidence was revealing to us about what is temperature. If physicists had succeeded in learning what temperature is from its empirical evidence, then, temperature, and its units of degrees, would have been defined, right from the start, using a combination of the same terms in which its evidence was expressed. The units of degrees are not expressible in terms of units that pre-existed them. There is no link between degrees and the units of the empirical evidence from which the existence of the property of temperature is inferred. That link must exist before we will learn what empirical evidence is revealing to us about what is temperature. In other words, we will not learn what temperature is until its units become defined units. Or, as I stated previously: It will be known when temperature has been explained because its units will become defined units. 

Nature does not provide us with scalars (Are you referring to pure numbers?) that are dimensionless. There is no such thing as a pure number. Every number has 'units'.. Those 'units' may be 'things'. 

Dr. Putnam: Thanks for your comments. As I said, we defer in our interpretation. I define the temperature as a derivative, but you want to measure it by a thermometer without defining it. In my opinion, that is all. May be, we should leave the things as they are.

The uninformative words primary and secondary have, in modern texts,replaced the informative words indefinable and definable. In order to make the basis for my discussion about temperature clearer for readers in general, I quote from:   

College Physics; Sears, Zemansky; 3rd ed.; 1960; Page 1, Chapter 1:

1-1 The fundamental indefinables of mechanics. Physics has been called the science of measurement. To quote from Lord Kelvin (1824-1907), "I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of Science, whatever the matter may be."

A definition of a quantity in physics must provide a set of rules for calculating it in terms of other quantities that can be measured.  Thus, when momentum is defined as the product of "mass" and "velocity," the rule for calculating momentum is contained within the  definition, and all that is necessary is to know how to measure mass and velocity. The definition of velocity is given in terms of length and time, but there are no simpler or more fundamental quantities in terms of which length and time may be  expressed. Length and time are two of the indefinables of mechanics. It has been found possible to express all the quantities of mechanics in terms of only three indefinables. The third may be taken to be "mass" or "force" with equal justification. We shall choose mass as the third indefinable of mechanics. 

In geometry, the fundamental indefinable is the "point." The geometer asks his disciple to build any picture of a point in his mind, provided the picture is consistent with what the geometer says about the point. In physics, the situation is not so subtle. Physicists from all over the world have international committees at whose meetings the rules of measurement of the indefinables are adopted. The rule for measuring an indefinable takes the place of a definition. ...

Chapter 15, page 286; 15-1:

To describe the equilibrium states of mechanical systems, as well as to study and predict the motions of rigid bodies and fluids, onlt three fundamental indefinables were needed: length, mass, and time. Every other physical quantity of importance in mechanics could be expressed in terms of these three indefinables., We come now, however, to a series of phenomena, called thermal effects or heat phenomena, which involve aspects that are essentially nonmechanical and which require for their description a fourth fundamental indefinable, the temperature. ...

Dear James,

This is a good question.

The zeroth law only involves thermal equilibrium, whereas, thermodynamic equilibrium involve thermal equilibrium, mechanical equilibrium, chemical and phase equilibrium. It is indefinable if we introducing the generalized force depend only on which “takes the same value in two systems” in mechanical equilibrium, or introducing the chemical potential depend only on which “takes the same value in two systems” in chemical and phase equilibrium, it was not true when we define the generalized force and the chemical potential,and so for temperature.

In fact, it is not an essential requirement, to measure temperature depend on “two systems in thermal equilibrium.” Such as physicists  measure the temperature of  the sun surface or a substance by optical thermometer.

I agree with your opinion, “I suggest that temperature is the same property without the need for separate treatment whether conditions are equilibrium or non-equilibrium. I further suggest that Clausius' thermodynamic entropy is the same property without the need for separate treatment whether conditions are equilibrium or non-equilibrium. These suggestions follow from my own theoretical work and from the continued condition that temperature is a fundamental indefinable property. The property is represented in physics equations only by its units. Its units of degrees are themselves indefinable units. It will be known when temperature is made a defined property because its units will, at that time, be made defined units. “

In my work, temperature is considered as the macroscopic energy level which represents the intensity of the heat energy. Where the heat energy denotes the energy of the thermal motion,  and is a state property,  which has not been well defined in current theories. We must define the heat energy first, as a state function, then we can consider its intensity  distribution --- the macroscopic energy level of the heat energy --- temperature T.

https://www.researchgate.net/publication/51990752_Internal_Heat_Energy_Entropy_and_The_Second_Law_A_New_Approach_toTheoretical_Thermodynamics

Article Internal Heat Energy, Entropy and The Second Law: A New Appr...

The quote from Sears and Zemansky clarifies what a definition is. For example, in f=ma force is defined in terms of the product of mass and acceleration. In other words, force is defined in terms of the three fundamental indefinable properties of mechanics. This equation cannot be solved for mass or length or time for the purpose of defining mass or length or time. Theoretical  physics adopted mass, length and time as being indefinable properties. Nothing can undo their indefinable status except to define them in terms of their pre-existing properties. Length and time have no pre-existing properties. They are permanently indefinable. In the case of mass, length and time are its pre-existing properties. A definition of mass can consist only of some combination of length and time. This circumstance clarifies what is the physics meaning of the word "defined."

This physics meaning of the word "defined" is the meaning used in my statement that all properties, with the exceptions of length and time, should be and could be defined in the same terms as their empirical evidence is expressed. All empirical evidence for all properties, other than length and time, consists only of combinations of measures of length and time. Velocity and change of velocity of objects are the two empirical combinations of measures of length and time. 

The question is what is temperature? We  will know what temperature is only after it has been made a defined property. Its definition will be some combination of measures of length and time. Physicists did not know and do not yet know how to form the definition of temperature. Without that definition, temperature remains unexplained. Physicists do not know what temperature is. My point in raising this question, other than making it clear that physicists do not know what temperature is, is to say that temperature can be defined. The solution of the definition of temperature follows automatically after mass is made a defined property. Both of these artificially indefinable properties represent lack of fundamental knowledge that infects the rest of theoretical physics.

I chose to focus on temperature rather than mass because physicists are absolutely convinced that they understand what mass is while they are not so absolutely convinced that they know what temperature is. Both properties should have been and could have been and can be made defined properties. This accomplishment would reform theoretical physics so that it takes its lead always from empirical evidence, and, no longer introduces unexplained, artificially indefinable, properties into physics equations.

The most beneficial change that will occur is that fundamental unity will be returned to the equations of physics right from the beginning of the reformed theoretical physics. That fundamental unity was first lost when it was decided that mass, or it could have been force, needed to be accepted as a fundamentally indefinable property. The decision to accept temperature as the fourth fundamental indefinable property ensured that "fixing" mass alone could not  restore theoretical physics' lost fundamental unity. We need to know: What is temperature?  

Hi Tang Suye,

Thank you for your message. I will read your referenced work in the next couple of weeks. I feel that I successfully made my point in print here. However, the discussion appears to have ended without evidence that the importance of it for theoretical physics is understood. The importance is that it is inevitable that there will be fundamental change. Blanks that have been tolerated, even ignored, must be filled in. 

Dear James,

It is an  interesting idea.  

The unit of absolute temperature is called the KELVIN *(K),  the physical meaning  of the temperature depend on

K=?/?.

Dear Tang,

This is the situation: Physicists have made errors at the fundamental level. The first error was when mass was made a fundamental indefinable property. The mistake would be just as serious if force had been made a fundamental indefinable property. The second error was when temperature was made a fundamental indefinable property. The third error was the formulation of a circular definition of electric charge. Its circular 'definition' is, of course, not a definition. It has served to misdirect attention away from recognizing that electric charge lacks a definition. The evidence that electric charge is actually a fifth fundamental indefinable property is that its units of coulomb's is not defined.  Since the units, meters and seconds, of empirical evidence are naturally and permanently fundamental indefinables, electric charge is a third artificial fundamental indefinable.  

The problem with accepting artificial indefinables into physics equations is that while the effect  upon understanding the nature of the universe is negative and greatly so, their presence will not prevent the equations from making good predictions. The theorists put their imaginations to work arriving at substitute properties and substitute explanations for those substitute properties. The meaning  of this previous sentence is that temperature is not what physicists say it is; mass is not what physicists say it is; and electric charge is not what physicists say it is. All three are introduced to us by empirical  evidence. It is the empirical evidence that can, if recognized, inform us as to what each of these properties are. This information has not be learned. Instead we have the results of theorists imaginations. They are free to imagine workable substitutes. The equations accept substitutes.

The evidence that this is the current situation is that the units of all three artificial indefinable properties were introduced as and remain as indefinable units. Those units cannot be defined until we recognize the information provided by empirical evidence that indicates to us how to define them. All three units, when defined, will be defined in terms of the units of empirical evidence only. This condition is what is required to meet the standard of a physics definition. 

What I am answering to  you is that T=?/? and M=?/? and e=?/?. The answers already exist on the internet. The overall result of putting these answers to work for us in the equations of physics is that fundamental unity appears constantly present in the reformed physics equations. This presence of fundamental unity does not leave room for theorists' imagined properties to be imposed into physics equations.  instead, all reformed properties, except the two properties of empirical evidence, are defined properties in the strict sense that is meant by 'physics definitions' as was explained by that Sears, Zemansky quote. There is no other proper way for properties, actually their units, to be introduced into physics equations. Any other attempt loses fundamental unity.

The benefit of fundamental unity is that there is a natural absence of multiple causes. While empirical evidence cannot, because of it consists of effects only, tell us what cause is, it can tell us that fundamental unity, having always been evidenced to exist by the orderliness of the universe, sources all effects to attributes of a single cause. When the equations of physics are returned to their natural, empirically defined forms, the plural is removed from 'fundamental force'. 

James A. Putnam

This thread appears to have run its course. I think this is an opportunity to propose an answer for those who understand we still are in need of one. The answer, with the following qualification, is that temperature is the rate of exchange of kinetic energy between molecules. For an ideal gas this is exact. For more complex arrangements  of energies, it needs further explanation. Now the qualification: The units of temperature, degrees Kelvin. are chosen to fit arbitrary conditions. This means that the measurement of temperature is not an accurate measurement of the rate of exchange of kinetic energy. The use of a proportionality constant whose magnitude is learned by first establishing the existence of an important fundamental increment of time, yields mathematical results that establish the correct form for temperature so that it correctly represents the rate of exchange of kinetic energy between molecules. if this possibility is not sufficient to encourage readers to read on, then I will throw in that you will learn the physical meanings of both Boltzmann's constant and the Universal Gas constant. 

Now it has been said what temperature is, and, what thermodynamic entropy is. I challenged others with the requirement that if they think that they know how to define temperature, then they should also show how they define the units of degrees Kelvin. Explaining the definition of a property is talk. In physics equations it is the units that represent the properties. Explanatory correctness depends upon correct units. So, I support what I have said about "What is temperature?" by giving its empirical units. Its empirical units are meters/second. The empirical units of thermodynamic entropy are seconds. Now I know that due to very strong resistance that these units will not be understood by probably all readers. So in the interest of pushing forward while being ignored. The empirical supported units of energy are meters. That is the reduced form. The full form is meters times the ratio of two accelerations. I mentioned the reduced form because the meters in temperature's "meters per second" represent energy and not distance. This is not so clear in the reduced form of units, but, it can be thought through in the full form of the units. Anyway, temperature, after supplying a proportionality constant to change normal temperature from its arbitrary scale to physical temperature, is the rate of transfer of energy between molecules.

Clausius' S=U/T, where U is heat and heat is energy in transit, shows that the empirical units of S are seconds. I offer the suggestion that Clausius' thermodynamic entropy is the time it takes for molecular kinetic energy to be transferred from molecule to molecule until it has been transferred to all molecules as if those molecules were lined up in a circle.Of course this needs further explaining and support, but, there is no professional outside interest in defining temperature let alone reading of a possible solution. However, the solutions learned from empirical evidence and not theorists go on. The Internet gets a thumbs up for freedom of speech. There is more to be said about what is thermodynamic entropy? Both Boltzmann's constant and the Universal Gas Constant are involved.  

I gave some direct answers. it is true that my message did not rigorously give the support for those answers. However, the support exists. I am addressing the meaning of Clausius' discovery of thermodynamic entropy. I offer two unfinished answers:

1. Boltzmann's constant, after applying a proportionality constant to correct for temperature's scale of degrees Kelvin, is the thermodynamic entropy of a single ideal gas atom, or even of a hydrogen atom. The Universal Gas Constant, after the same corrective measure is taken, is the thermodynamic entropy of a mole of ideal gas, or even of a mole of hydrogen gas.  

Obtaining the definition of temperature from the zero law seems to me to be a circular reasoning. In the zero law the concept of diathermic wall appears. But we define a diathermic wall as a wall that allows the passage of heat. And we define heat as energy in transit due to a temperature gradient. And here is the circular reasoning: to deduce the concept of temperature from the zero law we must first obtain the zero law which we obtain by knowing previously the concept of temperature.

I think it is better to leave the concept of temperature undefined as many others: space, time, position, etc.

A.H Wilson defines first an adiabatic wall as follows. If we have a general dynamical system with generalized coordinates a[i] and if the state of the system can only be changed by varying at least one of the a[i]s then the boundary of the system is adiabatic. For example the state of the system can be changed by moving the position of the wall (which varies the volume). As H.A. Buchdahl points out, this implies that the state of a system enclosed in an adiabatic boundary will be unaffected by changing the state of a second system separated from the first by an adiabatic wall. Any wall which is not adiabatic is diathermal. That is we can change the state of a system contained in a diathermal wall without moving the wall.

The point here is that the definition of diathermal and adiabatic are actually completely independent of heat. It is convenient to speak first to students in terms of heat transfer, which I actually do in my text. But then we must later carefully bring them to understand the difficulties in defining heat. Heat transfer may be defined (later) in terms of a temperature gradient, but that requires a definition of entropy production. That, of course, requires the Second Law.

Temperature is not an easy term for beginning students to understand. But the Zeroth Law does provide a definition of temperature as long as you use a careful (correct) definition of diathermal wall.

My own mentor J.R. Moszynski defined adiabatic and diathermal in terms of states of systems in equilibrium across adiabatic and diathermal walls, which is equivalent to Wilson and Buchdahl.

Carl S. Helrich

Sorry, I do not understand this definition, because the a[i] are undefined and how

to distinguish between heat- and power-exchange ? Perhaps a brief look to chap.

1.1 of my remarks in

Aspects of Non-Equilibrium Thermodynamics, World Scientific Singapore 1990

may help.

Best regards

W.M.

I think the term generalized coordinates is very well defined.The sum over the products of the generalized coordinates and the corresponding generalized forces is the work done. I cited other peoples' work to avoid self promotion.

With my best regards,

Carl Helrich

I am sorry for being short. The concepts of generalized forces and generalized coordinates are actually terms that originated in the analytical mechanics of Euler and Lagrange and finally Hamilton and Jacobi. After 1837 it was clear that any unity of ideas must be based on the general principles of analytical mechanics. Using the Einstein summation convention on repeated indices the infinitesimal work done in an infinitesimal process is then dW=da[i]A[i] where A[i] is the conjugate generalized force associated with an infinitesimal change on the coordinate a[i].

As William Thomson pointed out, after Joule presented the results of his work, Joule had accomplished this for a thermodynamic system. He had done work on an isolated system finding an equality between the work done and the change in state of the system. The fact that the generalized property measured in Joule's experiment was temperature is interesting, but incidental.

The issue is that work becomes the primary quantity and heat is the issue to be defined, and finally, with Gibbs, to be removed.

With my regards and hopes that these discussions help others. I think you and I understand the issue rather completely.

Carl Helrich

I have always thought that work, being a mechanical concept, must be defined first. Heat is then defined incorrectly by an appeal to the first law. Thus, I find Professor Helrich is making an important point. These concepts then can be applied to a nonequilibrium process also. One then finds that these quantities are not identical to exchange quantities commonly used in thermodynamics unless one is considering a reversible process. I have called them generalized work and generalized heat in my own writings on nonequilibrium statistical thermodynamics.

The concept of temperature is somewhat complicated. I follow Planck and believe that it must also be "defined" in a nonequilibrium state. So far, I have been able to define it for a restricted class of nonequilibrium states, which I call internal equilibrium states, in which the entropy is a state function of state variables; the latter now include regular observables like energy, volume, number of particles, etc. but also include internal variables required to specify a nonequilibrium state.

I believe have talked about it during the earlier discussion in this thread. This definition supercedes the equilibrium definition of temperature Professor Putnam is referring to.

Entropy being a state function allows one to write down the Gibbs fundamental relation from which the generalized heat dQ is identified as TdS; it looks almost identical to the equilibrium definition but my approach allows one to identify T as a nonequilibrium temperature of a system in internal equilibrium. I do not know if one can define T for any arbitrary nonequilibrium state but I hope that one can.

I strongly agree with Professor Helrich that thermodynamics is an extension of the classical mechanics; the only difference is that probability has to be be introduced there.

Hopefully, my comment would spur some discussion as it is an interesting issue.

Thank you Professor Gujrati. I find your point on the problem of nonequilibrium temperature important. I have always also made an assumption of local equilibrium to define a local temperature. But that is actually an ill-defined concept. Your approach in terms of restricted states sounds feasible.

One of the things I have always found attractive about thermodynamics is its subtlety. toward the end of his life Ed Jaynes called for a general information theory for nonequilibrium systems in order to gain an understanding of the Second Law. I agree with Jaynes.

Jaynes was a genious and his contributions were ahead of time. However, I have never understood if information theory is capable to "proving" the second law. In my view, the second law happens to be true but I do not know of any proof.

Please, have a look into the attached papers:

1: a non-equilibrium temperature can be defined in phenomenological thermodynamics

2: the same is true for a non-equilibrium entropy and an entropy production

3: discussions welcome

Professor Muschik has a very interesting approach to a phenomenological definition of temperature and I strongly urge those who are interested in the concept of temperature to read his contribution. Hopefully in near future various different notions of temperature will somehow merge together as our understnading improves.

Classical thermodynamics is a funny subject, no one really knows what temperature is, and what entropy is, because there are no explicit definitions which indicate the physical meanings of the two.

Classical thermodynamics itself cannot explain what is temperature, and what is entropy., we have only the equation, but there are no explicit definitions of the concepts.

“The concept of temperature. As a natural generalization of experience we introduce the postulate: if to assemblies are each in thermal equilibrium with a third assembly, they are in thermal equilibrium with each other. From this 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. This postulate of the ‘existence of temperature’ could with advantage be known as the zeroth law of thermodynamics.”[1]

However, this postulate of the ‘existence of temperature’ does not give the general definition of temperature, the postulate cannot explain what is temperature.

In classical thermodynamics, it is impossible to give an explicit definition which indicates the physical meaning of temperature.

[1]. Fowler, Ralph and Guggenheim, Eduard A. (1939). Statistical Thermodynamics: a Version of Statistical Mechanics for Students of Physics and Chemistry (zeroth law: pg. 56 - coined). Cambridge University Press.