What actually are microscopic and macroscopic viewpoints in thermodynamics?

Question

The microscopic viewpoint of studying a system in thermodynamics is the one in which we consider the system on a molecular/atomic/sub-atomic level. (is that even right?)

The macroscopic viewpoint is the one in which we ignore the molecular nature of the system and treat it as an aggregation of differential volumes, that have a limiting volume so that the system acts as a continuum.

If the above statements are true, then why temperature is considered a macroscopic concept?

Temperature is the measure of the average KE of the molecules of a system. Clearly, we're talking about molecules when we talk about temperature then why it is a macroscopic concept?

Answer 1

The question is about the meaning of terms such as 'microscopic' and 'macroscopic' in thermal physics generally.

To define these terms it is best to put to one side for a moment the idea that large things are often made of atoms and molecules and things like that. Rather, pretend you really don't know what sort of stuff the physical things under consideration are made of. Pretend you don't even know if they are made of continuous wobbly things like waves or itty bitty things like particles. So with this kind of ignorance, what can you still define, measure and reason about? The answer is: quantities such as volume and energy and mass and pressure (and electric field and viscosity and density etc. etc.). This is what thermodynamics is all about.

Now the question was, is temperature a macroscopic concept? The answer is yes it is macroscopic, because we can define it using macroscopic concepts. Temperature can be defined as that property which two system in thermal equilibrium with one another must have in common. This seems like a rather abstract definition at first, but stay with me for a moment. The next thing we need is a scale of temperature.

There are two ways to quantify temperature in thermodynamics. They are each equally profound, but not independent of each other, because either can be derived from the other. But note: neither of them mention kinetic energy!

The first way is to consider a pair of thermal reservoirs (also called 'heat bath') and imagine operating a reversible heat engine between them. In this case, suppose that when heat energy $Q_1$ is extracted from the first reservoir, heat energy $Q_2$ is delivered to the second reservoir. It is found that the ratio $Q_1/Q_2$ does not depend on what kind of process was involved, as long as it can be operated equally well forward (an engine) or back (a heat pump). (There is some very elegant reasoning that leads to this conclusion, starting from the laws of thermodynamics, but I am skipping that part). This leads to a way to define a temperature scale. The temperature scale is defined such that $$ \frac{T_1}{T_2} = \frac{Q_1}{Q_2}. \tag{1} $$ This is sufficient to define all temperatures because once you have a way to compare two temperatures, they can all be compared to some agreed case called unit temperature. Notice that nowhere in this argument is any mention needed of the microscopic composition of the systems involved. Nor did I need to introduce a microscopic description and then take a limit of large numbers.

The second way to define temperature in thermodynamics (as I said, equally fundamental as the one I just gave) is to use the expression: $$ T = \left. \frac{\partial U}{\partial S} \right|_{V, m,\, \rm etc.} \tag{2} $$ where $U$ is the entire internal energy of the system in question, $S$ is its entropy, and in the partial derivative properties of the system such as volume, mass and other things related to work are kept constant. (For readers unfamiliar with partial differentiation, I offer some simpler thoughts at the end. Here I am being completely precise and thorough). In order to use this second definition, we need to know what entropy is. One way to figure out what entropy is is to use the first temperature definition, plus some more clever reasoning named after Clausius, and eventually define entropy such that the second result holds. But you can if you like just assert that physical systems have a property called entropy, and assert some very general facts about it (e.g. it can only ever stay constant or increase in an isolated system), and then arrive at equation (2) as a definition of what we even mean by temperature (not just an assertion about temperature). In this approach it is normally felt to be more insightful to write it the other way up: $$ \frac{1}{T} \equiv \left. \frac{\partial S}{\partial U} \right|_{V, m,\, \rm etc.} \tag{3} $$ Equations (2) and (3) are saying precisely the same thing; I have just taken an inverse on both sides.

I have now shown that temperature is a macroscopic concept because I have only needed macroscopic physical ideas and quantities (energy, entropy, mass, volume) to define and describe it precisely.

It remains to say how temperature relates to microscopic behaviours and quantities. To find out the temperature of a collection of small things such as atoms or molecules or vibrations or whatever, the mathematical method amounts, in the end, to finding out the entropy and using equation (2) or (3). In many cases it turns out that the temperature is closely related to the mean kinetic energy of the parts of the system, but in order to say this in a quantitative way one has to be quite careful in deciding how the parts are being counted. But temperature is not a property of a single atom or a single vibration or a single rotation. It is a collective property, like an average. If atoms in a gas are moving around and colliding with one another, then at any given time some atoms will be moving fast, with lots of energy, and some will be slow, with little energy. But we should not say that in this case some atoms are hot and some cold. Rather, the temperature is a property of the distribution of energy. It is a measure of how quickly the number of atoms at a given energy falls off as a function of energy, when the atoms are continuously exchanging energy with one another through collisions. (This measure is somewhat related to the average energy per particle but they are not quite the same.)

Relating temperature to energy

Here is a further comment on the relationship between temperature and energy, suitable for school-level study. For many simple systems it happens that the entropy goes up in proportion to the logarithm of the energy, as long as the temperature is high enough: $$ S \propto \log U $$ with a proportionality constant of order $R$ (the gas constant): $$ S \simeq R \log U . $$ This implies that the energy is proportional to the exponential of the entropy: $$ U = A e^{S/R} $$ where $A$ is a constant. In this case $$ \frac{dU}{dS} = \frac{1}{R} A e^S = \frac{U}{R} $$ so using equation (2) we find $$ T = \frac{U}{R}. $$ This results works for many gases and solids at room temperature, as long as you understand I have omitted a factor of order 1 which depends on the individual case. The purpose of this added note is to show that temperature does often indicate energy, but it does not have to be like that. It happens when the relationship between entropy and energy is logarithmic, and this in turns happens when the dominant energy is kinetic (or potential in a harmonic well), and the system is excited well above its ground state.

Answer 2

A full answer to your question is simply reading a book on statistical physics.

The thermodynamic limit

There's no microscopic thermodynamics. At the microscopic level, you simply use ordinary mechanics (classical or quantum) for the particles.

But as the number of particles grow, this approach becomes both inefficient and impractical. So the trick is to resort to statistical analysis. This approach throws out a lot of information, since knowledge about individual particles is lost, but it matches empirical observation that this knowledge isn't necessary at the precision level required for macroscopic physics.

The limit $N\to+\infty$ with $N$ the number of particles is called the thermodynamic limit. This is where thermodynamics emerges.

Temperature

As for temperature (as well as other notions like pressure or interal energy), its definition makes it undefined at the microscopic level, so it has meaning only in thermodynamics (i.e. at the macroscopic scale).

In a nutshell, temperature is defined as the statistical average value of the kinetic energies of the particles. While an average value can always be computed, no matter the number of particles, standard deviation is another matter.

It's only with a high number of particles that standard deviation goes to zero, meaning that the average value becomes "almost certain" and represents the system correctly. That's when this value is called temperature.

As the microscopic level, temperature is undefined and unnessary, since we can work directly with kinetic energy.

Answer 3

Temperature is the measure of the average KE of the molecules of a system. Clearly, we're talking about molecules when we talk about temperature then why it is a macroscopic concept?

One should distinguish Thermodynamics and Statistical Physics.

• Thermodynamics is phenomenological macroscopic theory, describing complex systems in terms of parameters like temperature, volume, pressure, etc.
• Statistical physics is a microscopic theory that explains the same phenomena in terms of basic (quantum or classical) mechanics laws.

Thus, quantities like temperature, internal energy, entropy, etc. have different definitions in thermodynamics and statistical physics, which can be shown to be equivalent in terms of observable behavior. See here regarding different definitions of entropy.

Remark Although some textbooks (e.g., Huang, if I am not mistaken) choose to explain in parallel thermodynamics and statistical physics, many authors apparently find such an approach cumbersome and freely switch between the two, which produces confusion about the exact definitions of quantities (notably L&L, which is a very popular and authoritative text.)

Answer 4

I am not a physicist, and I won't give a technically precise answer, but I think a simple question deserves a simple answer, even if it leaves a lot more to be said.

I think the essence of your question is: if temperature is a measure of the average kinetic energy of individual particles in the system, how can temperature be a macroscopic quantity? The simple answer is that this is how statistics works. In a statistical description of a system one works with quantities that describe an aspect of the whole system, or at least a substantial part of it, which although in principle related to properties of all or elements of the system (or of that part) are very far from determining those properties. For instance in economics one might consider the average income of a population; although in principle that depends on the income of each and every individual, that is seldom the way it is actually determined (which more likely involves some clever sampling), and in any case is insufficient to determine, or indeed say much at all about, the income of any specific member of the population. The point is that an economist can work with the average income (any possible some more refined statistics) for her purposes (such as economic modelling), without ever directly considering individuals in the population.

Similarly in weather forecasting one works with pressures, temperatures, levels of humidity and whatnot, evaluated at some spatio-temporal grid of points. Each of these quantities can in theory be expressed in terms of the collection of all particles in the cell defined by the grid point, but that is not how its value is obtained; in practice only a measurement at the grid point is made, which involves in interaction with only a minute fraction of those particles. However, by the law of large numbers and under the hypothesis of a local state of equilibrium will, the value obtained will be a good approximation of the intended value. The model used for forecasting voluntarily only works with these values that are both statistical in nature and only approximatively known, because such simplifications make the model computationally feasible.

The macroscopic point of view does not deny the existence of individual particles, but accepts that information about their individual whereabouts is inaccessible, and is framed in terms of quantities (like temperature) that only represent statistics over vast amounts of particles, and moreover reflect those statistics up to a finite degree of precision.

Answer 5

There is no meaningful concept of micro vs. macro in thermodynamics. There is no natural scale to the universe. Consider this in relation to Einstain's claim that there is no absolute reference frame. Take this to its endpoint, then atoms are strictly a phenomenon of our existence with life, not essential to the universe itself.

The concept of entropy transcends the atomic model and goes straight to the source: energy (versus information). You don't need atoms to have mass, for example, only entanglements.

The issue of temperature, I claim, doesn't belong in physics, only pressure, volume, and phase.