Wednesday, May 29, 2013

What does "heating" mean at the nanoscale?

I've talked before about what physicists mean when they talk about "temperature", and work has me thinking about this a lot these days. Temperature is inherently a statistical concept.  It doesn't really make sense to talk about the temperature of a single electron.  The electron has some momentum (and therefore some kinetic energy), but temperature is not a meaningful concept for a single particle in isolation.  Now, if you have a whole bunch of electrons, you can talk about how many of them have a certain amount of energy.  That distribution of electrons as a function of energy takes on a particular form when the electron system is in thermal equilibrium.  (That is, if the electron system is weakly coupled somehow to an energy reservoir so that energy can be exchanged freely between the reservoir and the electrons.)  When the electron system is in thermal equilibrium with the reservoir, on average no net energy is transferred as a function of time between the electrons and the reservoir; this is what we mean when we say that the electrons and the reservoir have the same temperature.

The situation gets really tricky when a system is driven out of equilibrium.  For example, you can use a battery to drive electrons through some nanoscale system.  When you do that, and you look at different points within the nanoscale system, you will find that, in general, the distribution of the electrons as a function of energy doesn't necessarily look much like the thermal equilibrium case.  So, is there a sensible way to generalize the idea of temperature to quantify how "hot" the electrons are?  The problem is, there are many ways you might want to do this - you are trying to take a potentially very complicated distribution function and essentially summarize it by a single number, some local effective temperature.   A natural direction to go is to consider a thought experiment:  what if you took a reservoir with a well defined equilibrium temperature, and allowed it to exchange energy with the nonequilibrium system at a location of interest.  What reservoir temperature would you have to pick so that there is no net average energy transfer between the system and the reservoir in steady state?   That is one sensible way to go, but in the nano limit the situation can be very tricky, even in the thought experiment.  The details of how the imagined energy exchange takes place can affect the answers you get.  Tough stuff.


Anonymous said...

A related temperature question: When you calculate temperature fundamentally it is the thing that becomes equal at the most probable distribution (the log of the number of states divided by energy I think). But of course there is no way to measure that. Measured temperature is based on arbitrary reference points (melting, boiling, triple points, etc). Is it true that these are not directly connected?

Anonymous said...

I think that even "temperature" is a meaningful concept for very small systems. However, it makes no sense to estimate the temperature without estimating its uncertainty in small systems. Both numbers are of significance in order to make a reasonable physical statement.

Vipin said...
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Vipin said...

It seems to me that temperature cannot be defined in isolation without invoking the system's volume and particle number (Helmholtz energy), or pressure and particle number (Gibbs energy), or volume and chemical potential (free energy). My understanding is that the fundamental thermodynamic relation dU = TdS - pdV + N\mu sort of gives more meaning to the various quantities, as I believe your post seems to suggest while referring to the distribution of the particles' energies.

For instance, as far as I understand, even in thermal equilibrium there are temperature fluctuations given by dT^2 = T^2/C_v, with C_v being the specific heat calculated or observed at the mean temperature T.

Austen said...


It's not the specific heat that appears in that formula (which is an intensive quantity), but the heat capacity, an extensive quantity proportional to the amount of stuff. Thus in a large system the temperature fluctuations in the microcanonical ensemble (where the formula applies) are small.

Douglas Natelson said...

Anon@9:32, when I teach thermo/stat mech, I point out that you can't go and buy an entropy-meter. However, you can infer the entropy by integrating the heat capacity. The issue of truly absolute temperature standards is very interesting. See here:
There are measurements that are believed to be very close to absolute, primary thermometry, like Johnson noise.

Anon@5;52, yes, as Vipin points out (and the correction by Austen), you have to worry about temperature fluctuations as systems get small. However, I maintain that it makes no sense to talk about temperature of, e.g., a single particle at some instant in time. Vipin, I was coming from more the stat mech direction, but I agree with you.