For this answer, I'm going to extend the ambit of the question from 'laws of thermodynamics' to 'ideas from thermodynamics'.
The full answer to that broader question is surprising, fascinating and genuinely more interesting - as a plus, it answers this question too.tl;dr The laws of thermodynamics do not strictly apply at the quantum or classical microscopic level - but you can combine quantum mechanics and statistical mechanics to arrive at something remarkably satisfactory albeit incomplete.
[h=2]A Brief Primer on Thermodynamics[/h]
First, let's talk a little about thermodynamics itself. The 'laws of thermodynamics' is a vague term, since even in traditional thermodynamics what you mean by 'law' depends on context.
Historically, there have always been two major branches of the subject.
- There is, on one hand, the idea of heat and temperature applied to large macroscopic bodies, rooted in the study of heat engines, energy conservation and reversible vs. irreversible reactions. This is the famous thermodynamics, the one most useful to engineers and from where the ideal gas equation springs forth. Here, systems are characterised by states - certain parameters are presumed to pervade the system, and changes in any one constitute a new state for the system.
This, properly speaking, is what we usually mean by 'thermodynamics'. Its true name, however, is thermal physics.
- There is then the other thermodynamics, which seeks to move past generic desiscriptions of large macroscopic bodies and write abstract thermodynamical quantities in terms of physically real phenomena. Temperature becomes the statistical average of particle energy; pressure the statistical average of molecular collisions; and large macroscopic systems can now be described in terms of the averages of the states of its constituents.
This branch of thermodynamics is called statistical mechanics, and it is the backbone of all we study in thermodynamics.
[h=2]Why This Distinction is Needed[/h]
Interestingly enough, the three laws of thermodynamics
do not make sense or hold when it comes to statistical mechanics.
Oh, energy is definitely conserved, but the
definition of entropy changes radically enough that you
can see it
decrease spontaneously [FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]2[/FONT], and the third law of thermodynamics (that entropy is zero at zero Kelvin)
does not hold for imperfect systems like glass or carbon monoxide. The zeroth law doesn't even make sense when you think in terms of particle collisions.
And all of this is
without bringing quantum mechanics into the mix! This all comes of studying systems
classically or semi-classically (where constraints from quantum mechanics on the possible states of a system are taken into account) - typically, this is the sort of material you will find being covered in undergraduate physics lectures. So you see that asking whether the laws of thermodynamics carry over into quantum mechanics is problematic: they don't even carry over cleanly in
classical physics!
A much better question, then, remains:
while ideas from regular thermodynamics need not carry over perfectly, can ideas from statistical mechanics be made compatible with quantum mechanics?
[h=2]Integrating Quantum Mechanics with Statistical Mechanics[/h]
The answer is a resounding
yes, but not without changes. These changes may seem superficial at first glance, but are actually quite deep.
The application of ideas from thermodynamics to quantum systems is known as
quantum statistical mechanics. This is the foundation of modern statistical mechanics, and its roots go back to von Neumann's era.
Quantum mechanics' core idea is the replacement of
properties with
operators: instead of saying that energy is a property of an object, it says that there is an associated
function with the object which, when you apply a mathematical operation to it (hence
operator), returns a set of possible values.
A very naive way to begin to do quantum statistical mechanics is to
replace properties with their corresponding operators. So, instead of using
energy in the expression for the
canonical ensemble, you use the
Hamiltonian operator, and continue this practice everywhere. But this is just the start.
Some more ideas that are modified in the transition:
- Instead of using distributions (like the Boltzmann or Fermi-Dirac distribution) of properties, physicists now use density matrices - matrices that describe a mixture of different quantum states.
- Instead of the traditional statistical mechanics version of entropy:
we use instead the
Von Neumann entropy, which is meant to be used over the density matrix of a system:
- The time evolution of the system in question is now derived directly from Schrodinger's equation, rather than through complicated particle interactions.
These may
seem like simple analogous replacements of conventional statistical mechanics, but they hide an important and significant distinction:
everything now obeys the postulates of quantum mechanics. Every particle has been replaced by its wave-function, and our task, in quantum statistical mechanics, is to solve the really, really hard
many-body Schrödinger equation for the behaviour of the system. It's a fundamentally different way of dealing with and looking at things [FONT=MathJax_Main]3[/FONT].
[h=2]An Open Challenge[/h]
We now have a generally good framework for at least probing questions in statistical mechanics of a quantum nature. But are there aspects of quantum mechanics that are
not seemingly reconcilable with the nature of thermodynamics?
Yes. Quantum mechanics features only
unitary time evolution - what we usually take to mean
reversible changes - so a theory of statistical mechanics (which has to allow for
irreversible changes to a system - you can't turn glass back into the sand it used to be, can you?) would seem to be incomplete: unable to reproduce all of the reality we see. As a corollary, the second law of thermodynamics
cannot be derived in quantum statistical mechanics - for all intents and purposes, we cannot recover irreversible changes in a system at the quantum level.
This issue is explored (but not completely resolved) in immense depth in
Christian Gogolin's review paper on the foundations of quantum statistical mechanics, and has been tackled by giants like Schrodinger and von Neumann themselves. It remains an outstanding theoretical problem to obtain true irreversibility in quantum statistical mechanics, and we have not come too close to solving it. At best, we can arrive at pseudo-Second Laws - not the whole thing.
[h=2]Summary[/h]
This has been a general overview of the state of quantum statistical mechanics today.
Yes, there are aspects of thermodynamics that are irreconcilable with quantum mechanics, namely that irreversible processes for systems left to themselves
do occur in nature but not in quantum mechanics. Reproducing the second law of thermodynamics from quantum statistical mechanics is still a difficult challenge theoretically, and has not been accomplished.
[1]
Mark Eichenlaub's radical ideas about entropy, however, can salvage this situation somewhat.
[2] This is not to say that the second law in thermal physics (in other words, for macroscopic bodies) cannot be
recovered from statistical mechanics -
it definitely can. But in statistical mechanics, it only holds
on average - systems mostly tend towards increased entropy, but there are times where the entropy
can decrease. This is the gist of Lubos Motl's answer on Physics StackExchange (which I've linked to).
[3] Quantum statistical mechanics is an intensely theoretical field, closer to mathematics than physics, if anything. As a consequence, I am not actually aware of situations where quantum statistical mechanics provides better results than traditional statistical mechanics - I would be greatly interested in such an example, and I'll thank anyone who has one to share it with me.
