## Monday, December 13, 2010

### Temperature, thermal equilibrium, and nanoscale systems

In preparation for a post about a new paper from my group, I realized that it will be easier to explain why the result is cool if I first write a bit about temperature and thermal equilibrium in nanoscale systems.  I've tried to write about temperature before, and in hindsight I think I could have done better.  We all have a reasonably good intuition for what temperature means on the macroscopic scale:  temperature tells us which way heat flows when two systems are brought into "thermal contact".  A cool coin brought into contact with my warm hand will get warmer (its temperature will increase) as my hand cools down (its temperature will locally decrease).  Thermal contact here means that the two objects can exchange energy with each other via microscopic degrees of freedom, such as the vibrational jiggling of the atoms in a solid, or the particular energy levels occupied by the electrons in a metal.  (This is in contrast to energy in macroscopic degrees of freedom, such as the kinetic energy of the overall motion of the coin, or the potential energy of the coin in the gravitational field of the earth.)

We can turn that around, and try to use temperature as a single number to describe how much energy is distributed in the (microscopic) degrees of freedom.  This is not always a good strategy.  In the coin I was using as an example, you can conceive of many ways to distribute vibrational energy.  Number all the atoms in the coin, and have the even numbered atoms moving to the right and the odd numbered atoms moving to the left at some speed at a given instant.  That certainly would have a bunch of energy tied up in vibrational motion.  However, that weird and highly artificial arrangement of atomic motion is not what one would expect in thermal equilibrium. Likewise, you could imagine looking at all the electronic energy levels possible for the electrons in the coin, and popping every third electron each up to some high unoccupied energy level.   That distribution of energy in the electrons is allowed, but not the sort of thing that would be common in thermal equilibrium.  There are certain vibrational and electronic distributions of energy that are expected in thermal equilibrium (when the system has sat long enough that it has reached steady-state as far as its statistical properties are concerned).

How long does it take a system to reach thermal equilibrium?  That depends on the system, and this is where nanoscale systems can be particularly interesting.  For example, there is some characteristic timescale for electrons to scatter off each other and redistribute energy.  If you could directly dump in electrons with an energy 1 eV (one electron volt) above the highest occupied electronic level of a piece of metal, it would take time, probably tens of femtoseconds, before those electrons redistributed their energy by sharing it with the other electrons.  During that time period, those energetic electrons can actually travel rather far.  A typical (classical) electron velocity in a metal is around 106 m/s, meaning that the electrons could travel tens of nanometers before losing their energy to their surroundings.  The scattering processes that transfer energy from electrons into the vibrations of the atoms can be considerably slower than that!

The take-home messages:
1) It takes time for electrons and vibrations arrive at a thermal distribution of energy described by a single temperature number.
2) During that time, electrons and vibrations can have energy distributed in a way that can be complicated and very different from thermal distributions.
3) Electrons can travel quite far during that time, meaning that it's comparatively easy for nanoscale systems to have very non-thermal energy distributions, if driven somehow out of thermal equilibrium.

More tomorrow.

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