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1. Lecture 8: Fundamentals of Statistical Mechanics

   Schroeder Ch. 2 Gould and Tobochnik Ch. 4.1 – 4.4
2. Outline Classical and quantum systems Statistical ensembles Interacting thermodynamic systems Irreversibility Microscopic view of entropy and the 2nd law The 3rd law of thermodynamics
3. Introduction The essential methodology of statistical mechanics can be summarized as follows: Specify the macrostate and accessible microstates of the system Choose the statistical ensemble that is appropriate for the system Determine the mean values and other statistical properties of the system. In the following section, we will discuss the fundamental principles that address this methodology.
4. Formulation of the Statistical Problem The state of the system is the set of all variables one needs to describe that system at any given time. The macrostate of a system is the set of all macroscopic variables (e.g. pressure, volume, temperature, etc.) The microstate of a system is the set of all variables of the system required for a complete mechanical description of the system (e.g. positions and momenta of each particle) The multiplicity is the number of microstates in a given macrostate.
5. Classical Systems In order to fully describe the classical microstate of a system of particles, we need to specify all the coordinates and momenta of the particles of the system. We can obtain a countable number of microstates by assuming that the momenta and position of the system within a cell of area is given by This is sometimes called the continuum approximation
6. Classical Systems To find the multiplicity, we must sum over all the cells that lie within the region of phase space. Thus Two common examples of classical systems are Classical ideal gas Classical harmonic oscillator
7. Classical Ideal Gas Consider a gas of non-interacting particles in a box and suppose that we also know the total internal energy of the system. Because the energy of the system is simply the kinetic energy of the particles, we have If the system consists of identical particles of mass , we could write the equation as
8. Classical Ideal Gas The multiplicity of the system is given by It can be shown that the surface area of the dimensional momentum sphere is given by Thus, the multiplicity of the system gives
9. Classical Harmonic Oscillator Consider a system of identical, non-interacting harmonic oscillators. If we know the total energy of the oscillators, then Performing the integral over phase space gives
10. Quantum Systems The quantum states of particles are usually defined by a set of quantum numbers. A microstate of a quantum system is completely given once its wavefunction is known and the wavefunction is given by its quantum number We will consider the following quantum systems Spin ½ paramagnet Einstein solid
11. Spin ½ Paramagnet A paramagnet is a material in which the constituent particles (called magnetic dipoles) tend to align parallel to any external applied magnetic field. For any microscopic dipole, quantum mechanics only allows discrete values for the component of the dipole moment vector along any given axis. Consider a paramagnetic system consisting of non-interacting spin-1/2 particles, each possessing a magnetic moment , placed in an external magnetic field .
12. Spin ½ Paramagnet A particle that is parallel (antiparallel) to the magnetic field has energy The internal energy of the particles will depend on the orientation of the particles. For a spin ½ paramagnet, the multiplicity of any macrostate is the same as the multiplicity of the binomial distribution
13. Einstein Model In 1907, Einstein proposed a model that reasonably predicted the thermal behavior of crystalline solids. In this model, a crystalline solid containing atoms behaves as if it contained independent quantum harmonic oscillators. A quantum harmonic oscillator has discrete energy levels given by
14. Einstein Model To describe a macrostate of an Einstein solid, we have to specify and . To describe a microstate of an Einstein solid, we have to specify for oscillators. Let’s start with a very small Einstein solid containing only three oscillators (e.g. one atom).
15. Einstein Model How many microstates exist for a given macrostate with oscillators and total energy ? An equivalent question to ask is: How many ways can identical objects be distributed into identical boxes? Since each quanta must end up in an oscillator, the last object of the distribution must be an oscillator. Therefore, the multiplicity of a state of oscillators ( atoms) with energy quanta distributed among these oscillators is given by:
16. Statistical Ensemble How can we determine the probability in which a system finds itself in a given microstate at any given time? We can perform the same experiment simultaneous on a large number of identical systems called an ensemble. If there are such identical systems and state is found times, then the probability that state occurs in any experiment is
17. Statistical Ensemble Suppose we are interested in knowing about some property of the system . Since we cannot know the precise microstate the system is in, we generally ask for and . Based on the central limit theorem, when the system is sufficiently large, statistical fluctuations must be small compared with its average value We will study three statistical ensembles in this course The microcanonical ensemble (isolated systems) The canonical ensemble (closed systems) The grand canonical ensemble (open systems)
18. The Fundamental Postulate of Statistical Mechanics In order to use statistical ensembles to determine the probability of a system being in a given microstate, we need to make further assumptions to make further progress. A relatively simple assumption for a wide variety of thermodynamic systems is the assumption of equal a priori probabilities: In an isolated system in equilibrium, all accessible microstates are equally probable
19. Interacting Thermodynamic Systems Now that we have properly formulated the statistical problem, we can now use the methodology of statistical mechanics to examine thermodynamic systems. We will now examine the energy exchange between two thermodynamic systems by considering Two interacting Einstein solids Two interacting ideal gases
20. Interacting Einstein Solids Consider two Einstein solids that can exchange energy with each other and remain isolated from the rest of the universe. To specify the macrostate, we must specify the individual value of and . The table below shows the results for and
21. Interacting Einstein Solids The table below shows the results for , and Note that the most likely macrostate is much more likely than the least likely microstate
22. Interacting Einstein Solids Suppose that this system is initially in a state with . As the system approaches thermal equilibrium, it is overwhelmingly probable that energy has followed from B to A. Thus, the system has exhibited irreversible behavior.
23. Implications of Irreversibility If the system is not in the most probable macrostate, it will rapidly and inevitably move toward that macrostate because there are far more microstates in that direction. This is why energy flows from “hot” to “cold” and not vice versa (the Clausiusstatement). The system will subsequently stay at that macrostate(or very near to it), in spite of the random fluctuations of energy back and forth between the two solids. When two solids are in thermal equilibrium with each other, completely random and reversible microscopic processes tend at the macroscopic level to push the solids inevitably toward an equilibrium macrostate.
24. Multiplicity and the 2nd Law The spontaneous flow of energy stops when a system is at, or very near, its most likely macrostate (i.e. the macrostate with the greatest multiplicity). Thus, the 2nd law of thermodynamics can be understood as a law of increasing multiplicity. As becomes large, then only a tiny fraction are reasonably probable and the multiplication function becomes very sharp. We will demonstrate this for the Einstein solid and the ideal gas.
25. Multiplicity of a Large Einstein Solid The multiplicity of an Einstein solid with oscillators ( atoms) and energy quanta distributed among these oscillators is given by: Using Stirling’s approximation, it can be shown that the multiplicity of an Einstein solid with (the “high temperature limit”) is For a system of two large Einstein solids, the multiplicity of the combined system is given as
26. Interacting Large Einstein Solids The height of the peak is given by It can be shown that the multiplicity of the combined system is
27. Interacting Large Einstein Solids When two large Einstein solids are in thermal equilibrium with each other, any random fluctuations away from the most likely macrostate will be unmeasurable. The limit where a system becomes large enough so that the measurable fluctuations away from the most likely macrostate never occur is called the thermodynamic limit.
28. Multiplicity of an Monatomic Ideal Gas As shown previously, the multiplicity of an ideal gas is a function of , , and The above expression can be written in separable form
29. Interacting Ideal Gases Suppose now that we have two ideal gases, separated by a partition that allows energy to pass through. If each gas has molecules (of the same species), then the total multiplicity of this system is
30. Interacting Ideal Gases The multiplicity function has a sharp peak with a peak width of If we allow the gases to exchange volume, then the multiplicity function has a sharp peak with a peak width of
31. Multiplicity and the 2nd Law The previous examples demonstrate that for any thermodynamic system Energy tend to rearrange themselves until the multiplicity is near its maximum value. Any large system in equilibrium will be found in the macrostate with the greatest multiplicity Multiplicity tends to increase All of these statements are restatements of the second law of thermodynamics.
32. Entropy and Multiplicity Consider an isolated system of 4 particles divided into a subsystem. Assume that the system is thermally insulated from its environment and that each subsystem is insulated from each other. The total number of microstates accessible to the system is
33. Entropy and Multiplicity Suppose that the insulating partition separating the subsystems is changed to a conducting partition. The total number of microstates accessible to the isolated composite system is
34. Entropy and Multiplicity Note that the multiplicity of the composite system increased when the internal constraint is removed. By the 2nd law of thermodynamics, there must be an increase of entropy as energy is exchanged between both subsystems. This implies that there is a relationship between entropy and the multiplicity of a given macrostate.
35. Entropy and Multiplicity Based on these observations, Boltzmann defined the entropy as Thus, entropy is a function of the number of accessible microstates of a given system. Since small (large) multiplicity implies order (disorder), this implies that an increase of entropy implies an increase of disorder. This implies that spontaneous processes always occur because of a net increase of entropy.
36. Entropy and the 2nd Law An isolated system, being initially in a non-equilibrium state, will evolve from macrostateswith lower multiplicity (lower probability, lower entropy) to macrostateswith higher multiplicity (higher probability, higher entropy). Once the system reaches the macrostateswith the highest multiplicity (highest entropy), it will stay there. Therefore, the entropy of an isolated system never decreases. Thus, any process that increases the number of microstates will happen (if it’s allowed by the 1st law).
37. Entropy, Temperature, and Internal Energy Recall that the relationship between entropy and temperature is given by Now we can get obtain an energy equation of state for any system for which we have an explicit formula for the multiplicity. Thus, we’ve bridged the gap between statistical mechanics and thermodynamics.
38. Entropy, Temperature, and Internal Energy The procedure is Find for a given thermodynamic system Evaluate the entropy using Evaluate the temperature using Solve for
39. High Temperature Einstein Solid – Example For a high temperature Einstein solid, the multiplicity is given by The entropy is given by The temperature is given by Therefore, the internal energy is given by
40. Low Temperature Einstein Solid – Example For a low temperature Einstein solid, it can be shown that the internal energy is given by As , the energy goes to zero as expected
41. The 3rd Law of Thermodynamics The statement of the third law of thermodynamics was originally postulated by Walther Nernst and later modified by Planck At , the specific entropy (entropy per particle) is constant, independent of all the extensive properties of the system Since , the third law implies that at zero temperature, every isolated thermodynamic system should settle into its unique lowest-energy state called the ground state.
42. The 3rd Law of Thermodynamics Recall that the total change of entropy of a thermodynamic system is given by If , then the 3rd law of thermodynamics implies that The entropy associated with a system’s ground state is called the residual entropy.
43. The 3rd Law of Thermodynamics The residual entropy occurs because It’s possible to change the orientation of molecules in some solid crystals with very little change in energy Mixing of different isotopes of an element Multiplicity of alignments of nuclear spins
44. The 3rd Law of Thermodynamics In order for , the integrand must converge, which implies that as . Thus, the third law says that the classical theory of heat capacities breaks down at low temperatures. We will show later in this course that at low temperatures