What is the difference between thermodynamics and statistical mechanics?
Thermodynamics – Differences Between Thermodynamics and Statistical Mechanics
statistical mechanicsthermodynamics
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Actually I think I disagree with the answer by BMS (the group of asymptotic symmetries of asymptotically flat spacetimes?). However I am not sure to have understood BMS'answer completely.
In my opinion, there is no difference between the definition of work in pure mechanics and work in thermodynamics (I stress that I am speaking of thermodynamics and not statistical mechanics). In both cases it is computed by the integral of ${\bf F} \cdot {\bf ds}$, taking all forces acting on the system into account. In the pure mechanical case the theorem of energy conservation says that $$W = \Delta U + \Delta K\:.\qquad (1)$$ $W$ is the work done on the system by external systems, $K$ its kinetic energy and $U$ the total potential energy of internal forces. When considering situations where the work $W'$ of the system on the external systems coincides, up to the sign, to the work $W$ done by the external system on the system (and this is not the case discussed by BMS) we can also say that: $$\Delta U + \Delta K + W' =0\:. \qquad (2)$$ In real physical systems, one has to consider the fact that a system receives energy also in terms of heat, $Q$: that is energy that cannot be described in terms of macroscopic work. In this case (1) has to be improved as $$W + Q = \Delta U + \Delta K\:.\qquad (3)\:.$$ Actually, also the definition of $U$ has to be improved in (3), since it has to encompass the thermodynamic internal energy in addition to all types of macroscopic potential energies.
Referring to standard system of thermodynamics (thermal machines), where $\Delta K$ is negligible and the work done by the external system is identical up to the sign to that done by the system, (3) simplifies to $$\Delta U = Q -W'\:,$$ that is the standard statement of the first principle of thermodynamics for elementary systems. However the general form is (3).
It is worth stressing that this picture needs a sharp distinction between macroscopic description (essentially done in terms of continuous body mechanics) and microscopic description, completely disregarded but embodied in the notions of heat and internal (thermodynamic) energy. If, instead one considers also the microscopic (molecular) structure of the physical systems, the distinction between work and heat is more difficult to understand since both are represented in terms of forces. Nevertheless exploiting the statistical approach to Hamiltonian mechanics the said distinction arises quite naturally.
Focusing on the system given by a rigid block discussed by BMS, the absolute value of the work $W$ done by the friction force acting to the block due to the ground (that eventually stops the block), is different from the absolute value of the work $W'$ done by the block on the ground. The former amounts to $W= -K$ the latter, instead, is $W' = 0$. The energy equation for the block is:
$$W + Q = \Delta U + \Delta K\:.$$
$Q$ is the non-mechanical energy entering the block during the process, responsible for the increase of its temperature. Since $W= -K$ one can simplify that equation to
$$Q= \Delta U\:.$$
The equation for the ground (for instance a table) is instead simply:
$$Q' = \Delta U'$$
Now $Q' \neq -Q$ and $W'=0 \neq -W$. The fact that $Q+Q' \neq 0$ it is important because it says that there is a heat source between the contact surfaces of the two bodies, and the total heat is not conserved (as conversely was supposed in the original theory of heat, the "flogisto" represented as a fluid verifying a conservation equation).
If referring to the overall system made of the block and the table, since no energy enters it, the equation is
$$\Delta U + \Delta U' + \Delta K =0\:.$$
That is
$$\Delta U + \Delta U' = -\Delta K >0\:.$$
It says that all the initial kinetic energy is finally transformed into internal energy producing the increase of temperature of both the block and the table.
I think this is a nice question! I hadn't appreciated before that you could describe both classical field theory and statistical mechanics as different kinds of "large $N$" limits of classical mechanics ($N$=number of degrees of freedom).
The TL;DR:
- Classical field theory involves an infinite number of classical degrees of freedom at zero temperature.
- Statistical mechanics involves a finite number of classical or quantum degrees of freedom at finite temperature.
- Statistical field theory involves an infinite number of classical or quantum degrees of freedom at finite temperature.
- Quantum field theory involves an infinite number of quantum degrees of freedom at zero temperature.
You will find examples which contradict essentially all of the distinctions I just made :) (You can compute finite temperature effects in quantum field theory, you will find statistical mechanics books covering the infinite number of degrees of freedom of the electromagnetic field at finite temperature, etc) But I think it captures what's going on at a high level, at least morally.
Classical field theory
A typical entry point into classical field theory is to consider an infinite number of coupled harmonic oscillators. In classical mechanics, you study a mass on a spring. In a more advanced treatment, you will study two or more coupled masses, for example two or three masses connected by springs. You can approach classical field theory by filling space with a dense grid, where at every node in the grid there is a mass, and each mass is connected by springs to its nearest neighbors. This "infinite box spring" can describe wavelike solutions that propagate through space if you "ping" one of the point masses. The questions you tend to ask are deterministic. Given some initial condition of the field, you want to know how the field will evolve. The answer to a given question will be based on the configuration of the field at a given time.
Statistical mechanics
The essence statistical mechanics is random motion due to thermal fluctuations that occur at finite temperature. You can imagine taking one mass on a spring, and coupling it to a large thermal bath at some temperature. If we focus on how the mass on a spring behaves, it will undergo random oscillations due to its connection to the thermal bath. The thermal bath is the large system, but we do not need to explicitly model it. Sometimes, people talk about "ensembles" of hypothetical copies of the original system which are coupled together. The questions you ask in statistical mechanics are probabilistic. You want to know how likely it is for the system to have some property. Answers to questions are based on a probability distribution over states of the system.
Statistical field theory
I want to make a few other points about systems displaying random motion.
Imagine a classical field, which is undergoing random motion. Imagine our infinite box spring, coupled to a thermal bath with some temperature. There will be random waves propagating through the mattress. This is statistical field theory. You can use this set of ideas to study states of matter, and in particular phase transitions between different states of matter. You can think of all of the atoms in a magnet, say, as describing a kind of field (imagine each atomic magnet as a point mass in our box spring analogy, and the springs as an analogy for magnetic couplings between the atoms). Then, depending on the temperature, the atomic magnets may line up to form a magnetized state, or be randomly oriented to be a demagnetized state -- statistical field theory lets you quantitatively study this kind of situation. Answers to questions are based on studying the probability distribution of field configurations. This is as opposed to classical field theory, where answers are based on a single, deterministic field configuration, and statistical mechanics, where the answers are based on a probability distribution of some finite number of degrees of freedom.
Quantum field theory
Quantum field theory is like statistical field theory, but where the random motion is driven by quantum mechanics, instead of temperature. A quantum harmonic oscillator exhibits random zero-point motion, and there is a discrete spectrum of excitations above this ground state. Coupling many quantum harmonic oscillators to form a "quantum boxspring mattress" is one way to think about quantum fields (this is the approach Zee takes in his textbook). The discrete spectrum of excitations for one quantum harmonic oscillator, can be interpreted as particles in quantum field theory. Answers to questions in this context are based on the probability amplitudes for the field to achieve certain configurations (a probability amplitude $A$ is related to a probability $P$ by $P=|A|^2$).
Caveats
Finally, I should say that the boundaries between different areas in physics are not always as well-defined as they may seem, especially between statistical mechanics and statistical field theory. In my experience, the factor that really separates statistical mechanics and statistical field theory is what tools are used to solve a problem, rather than whether a classical or quantum field is involved. Blackbody radiation is often covered in a statistical mechanics course, for example, even though you are really studying the statistical mechanics of the electromagnetic field. I think this is just considered "too basic" to be called statistical field theory, even though by my definition above, arguably it should be a topic in statistical field theory. To give you an idea, a central topic in a statistical field theory course is the Ginzburg-Landau approach to phase transitions, where one uses symmetry to motivate an expression for the free energy of a field, and different states of matter correspond to how the minima of the free energy depend on temperature. More generally, statistical field theory courses often use techniques that are also used in quantum field theory -- such as effective field theory, the renormalization group, non-perturbative techniques such as instantons -- but applied to thermal systems (not to say that the techniques originated in QFT; there is a rich interplay where information passes back and forth between the subjects). This is just to say; many of the divisions we make in physics are somewhat arbitrary, and so always take claims that different fields are cleanly separated (like I made in the preceding paragraphs of my answer) with a grain of salt.
Best Answer
Statistical Mechanics is the theory of the physical behaviour of macroscopic systems starting from a knowledge of the microscopic forces between the constituent particles.
The theory of the relations between various macroscopic observables such as temperature, volume, pressure, magnetization and polarization of a system is called thermodynamics.
first page from this good book