Chapter 18 Statistical mechanics of classical systems


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1 Chapter 18 Statistical mechanics of classical systems 18.1 The classical canonical partition function When quantum effects are not significant, we can approximate the behavior of a system using a classical representation. Here, we derive the expressions for the partition functions in such cases. We will consider a singlecomponent system composed of sphericallysymmetric, singleparticle molecules. For the canonical partition function, the discrete version in general is,,, = exp all at, all DOF = DOF for DOF for DOF for (18.1) where is an index running over all microstates of the system and the sums indicate exhaustive enumeration of all possible values for each degree of freedom (DOF) of each particle. For a classical system of structureless particles, each microstate involves 6 degrees of freedom the 3 positions and 3 momenta for each particle. These variables are continuous, rather than discrete. We rewrite the summation over microstates as an integral over these variables:,,~!",# "$ %! %! %! %# %# %# (18.2) Here, we have replaced the energy with the Hamiltonian function. The quantities # and! are the vectors of all positions and momenta, respectively. The notation %! indicates %& ' %& ( %& ) such that the integral over %! for one particle is actually a threedimensional integral. A similar case exists for %#. Therefore, the total number of integrals is 6. We say that this is a 6dimensional integral. We must first discuss the limits of integration: For each momentum variable, the limits are < & <. In other words, the momenta can take on any real value. These degrees of freedom and the associated integrals are unbounded. For each position variable, the values it can take on depend on the volume of the container into which it was placed. For a cubic box of side length,, one has 0 <. <,. Therefore these position variables are bounded and the integrals are 1
2 definite. The limits of the position integrals also introduce a dependence of the partition function on. Does the specific shape of the container matter? For large homogeneous systems, this integral is independent of the container shape, for the same average density. The reason is that any unusual container shape can be broken down and approximated by a number of smaller subsystems of the same density that are each cubic in size, in a pixilated fashion. Each pixel can still be large enough to be considered macroscopic in size, which means that boundary interactions between it and neighboring pixels are very small. Therefore, the net properties of the whole system are simply the sum of those of the small pixels, which is insensitive to how they are arranged (the overall shape). Notice that the classical partition function is not dimensionless, as it has units of /momentum5 6 /distance5 6 owing to the integrand variables (%! s and %# s). This means that we can never calculate absolute values of purely classical partition functions, since the value would change depending on what units we used. Instead, we can only compute ratios of partition functions, or differences in free energies, since these quantities are dimensionless. This result mirrors our considerations of the third law of thermodynamics, where we found that a classical description is inadequate for providing an absolute entropy. The general rule is that: In a purely classical description, where we have continuous degrees of freedom in the form of atomic positions and momenta, we cannot compute absolute values of partition functions. Instead, we can only compute ratios of a partition function at different states. In reality, we know that the world is not classical, but behaves quantummechanically. In the hightemperature limit, however, quantum systems behave like classical ones. Therefore, we typically introduce a correction factor into the classical canonical partition function to make it agree with the quantum behavior in the hightemperature limit. This correction factor looks like:,, = 1 h 6!!",# "$ %! %! %! %# %# %# (18.3) Here, h is simply Planck s constant, and! is the quantum correction for indistinguishable particles, which is the general case for nearly all of matter. Notice that the factor of h provides a natural metric that makes the canonical partition function dimensionless, since h has units of /momentum5 /distance5. This correction means that we can use the classical approximation at high temperatures to compute absolute partition functions. Since they must integrate to a value of one, the correction also applies to the microstate probabilities: 2
3 !,# = exp/ A!,# 5 h 6!,, (18.4) We can simplify the partition function further since we know that the total energy is the sum of potential and kinetic energies: This enables us to write: A!,# = B! +D# (18.5),, = 1 h 6! EF!"$ %! %! %! GE H #"$ %# %# %# G (18.6) Here, we are abbreviating the multidimensional integrals with a single integral sign for simplicity. We were able to make this separation of the kinetic and potential energy parts because the kinetic energy depends only on the momenta, and the potential energy only on the positions. Therefore, the exponentials can be factored using the properties of integrals that IJKL%J%L = E IJ%JG E KL%LG (18.7) With the separation above, we can simplify the kinetic energy term, since we know that B! = 1 2N Substituting into the kinetic energy integral: O & O,' + & O,( +& O,) $ (18.8) F!"$ %! %! %! = P RS T,U V V V T WS T,XWST,YZ %! %! %! = E P S V [,U %&,' G E P S V ",Y %&,) G ] = \ ] P SV %& 6 ^ (18.9) The last line in this equation comes from the fact that all of the 3 integrals are identical, for identical particles. We have also indicated the limits explicitly. The integral can be evaluated analytically. The result is: So, the total canonical partition function becomes: F!"$ %! %! %! = _ 2`N 6 a (18.10) 3
4 ,, = 1 _2`N 6! h a H # "$ %# %# %# (18.11) The combination of the constants in front is a familiar grouping for us. Recall, it is related to the thermal de Broglie wavelength, Λ = h 2`Nc d. Simplifying, we arrive at our final expression:,, = f,, Λ 6! where f H #"$ %# %# %# (18.12) Here we introduced a new variable f,,, which is called the configurational partition function. That is, f only depends on the positional part of the degrees of freedom and the potential energy function. All particles with the same mass will have identical contributions from their kinetic degrees of freedom (the momenta), but may have distinct modes of interacting which will result in different contributions from the configurational part. Importantly, The kinetic and potential energy contributions to the canonical partition function are separable. The kinetic contribution can be evaluated analytically, and for simple spherical particles, it gives the thermal de Broglie wavelength. Thus, the task of evaluating the properties of a classical system actually amounts to the task of evaluating the configurational partition function f. In general, this is a complicated task, since the multidimensional integral over the 3 positions is nontrivial to evaluate in the general case, owing to the complex functionality of typical potential energy functions. A significant body of statistical mechanics is devoted to developing good expansion and perturbation techniques that can be used to approximate the configurational integral. Note that in the special case of identical noninteracting classical particles, where the potential energy function is separable between all of the particles, one can write:,, = i! where i 1 Λ 6j# %# (18.13) where i is the singleparticle partition function and k gives the potential energy of a single particle. EXAMPLE 18.1 Show that the quantumcorrected classical partition function gives the correct free energy for an ideal gas. For an ideal gas, Dl = 0. Therefore, f,, = m %l %l %l = 4
5 The total canonical partition function is:,, = n 6! = _ a n 6 The second line makes use of Stirling s approximation. Using o = c d p, we find: o = c d p_ n 6 a c d This is the same result we found in the previous lectures on ideal gases, where we computed q = c d pr using quantum mechanics and used a Legendre transform to find o Microstate probabilities for continuous degrees of freedom The canonical partition function enables us to compute the probabilities of every microstate in the system. Therefore, we can compute all sorts of microscopic properties that were not accessible from a purely macroscopic perspective. Let s consider one particular microstate N. The microstate N is specified by a set of values for all of the classical degrees of freedom,! P,# P. The probability that the system will visit this microstate is given by:! P,# P =! s ",# s " $ t!",# " %! %! %! %# %# %# (18.14) There is a subtle difference between this microstate probability and what we obtained for a discrete system. Here, since the momenta and positions are continuous, we have to talk about differential probabilities. That is,! P,# P %! %! %# %# is proportional to the differential probability that the system is at a microstate that lies between &,' %&,' 2 and &,' + %&,' 2, &,( %&,( 2 and &,( + %&,( 2, and so on and so forth. In other words, the probability corresponds to a microstate within a differential element %! %! %# %# around the phase space point! P,# P. In general, for probabilities for a continuous function, J,L for continuous variables J,L is defined such that J,L%J%L gives the differential probability that J and L lie within an infinitesimally small region %J centered around J and %L centered around L. Therefore, J,L has dimensions of /J5 /L5 and J,L%J%L is dimensionless. In this sense, J,L is called a probability density since it gives probability per unit of J and L. Just as the sum of the probabilities of all microstates in a discrete system must equal one, the integral of all differential probabilities in a classical system must equal unity: 5
6 !,# %! %! %# %# = 1 (18.15) Here, the limits of the integrals are the same as those mentioned above for the canonical partition function. Returning to the probability for microstate N, we can rewrite the expression by separating into kinetic and potential energy contributions:! P,# P = F! s "$ H # s " $ t F!" H# " %! %! %! %# %# %# (18.16) Notice that we can factor the exponentials and the integrals in the same way that we did before, to obtain: F! s " $ H # s " $! P,# P = u vu v (18.17) t F!" %! %! %! t H#" %# %# %# Each of these two energy terms is itself a probability dependent on the Boltzmann factor, with a normalization in the denominator. Thus, we can treat a classical canonical system as composed of two subsystems of which one entails the kinetic degrees of freedom and the other the configurational degrees of freedom. Importantly, this equation suggests that the joint probability of one set of momenta and one configuration is given by the product of the two separate probabilities, The important concept is the following:!,# =! # (18.18) In the canonical ensemble, the configurational and kinetic contributions to the partition function are completely separable. The total partition function can be written as a product of partition functions for these two kinds of degrees of freedom. Moreover, the probability distribution of atomic configurations is independent of the distribution of momenta. The former depends only on the potential energy function, and the latter only on the kinetic energy function. If we are interested in only the distribution of the momenta, we can use the analytical integrals performed above for the kinetic energy to find that:! = F!" $ h 6 Λ 6 (18.19) This expression gives the probability with which a particular set of momenta for all atoms will be attained by the system at any one random instant in time. On the other hand, if we 6
7 are interested distribution of the atomic configurations, l, regardless of the set of momenta, the considerations above show that: # = H #" $ f,, (18.20) You can think of this expression as giving the probability that particular sets of # particular atomic configurations are visited by the system. Notice that nothing remains in this equation that has anything to do with the kinetic energy. That is, everything here depends on the potential energy. In other words, there is a complete separation of the probability distributions of momenta and of spatial configurations. These degrees of freedom do not interact with or influence each other. The likelihood of the system visiting a particular configuration of atomic positions is completely independent of the velocities of each atom in that configuration. Let s compare the two kinds of degrees of freedom side by side, writing out the kinetic distribution over all momenta: property kinetic configurational contribution to total partition function probability distribution F!"$ %! %! %!! F!" $ = t F!" %! %! %! H #"$ %# %# %# # H #" $ = t H#" %# %# %# 18.3 The MaxwellBoltzmann distribution Here, we will find the distribution of velocities of particles at a given temperature. This distribution is called the MaxwellBoltzmann distribution. From above, we have:! = F!" $ h 6 Λ 6 (18.21) Let s focus on particle number 1 and one component of its momentum, &,'. We want to find just the distribution of &,', not the joint distribution of all of the momenta. We have to think of this as finding the probability that we will see a particular value of &,' regardless of the values of momenta of all the other components and particles. How do we do this? Well, if we were going to find the probability that &,' has a value of zero, we would need to enumerate all of the microstates where &,' = 0 and sum up their 7
8 probabilities. Here, though, our degrees of freedom are continuous and therefore the sum corresponds to an integral. Therefore, we can say in general that, Given a joint probability density J,L,w, the probability density of a specific value of J, regardless of the values of L and w, is given by integrating the distribution over the variables not of interest, J = tt J,L,w%L%w. As expected, the dimensions of J are /J5 since the dimensions of J,L,w are /J5 /L5 /w5 and the integral adds dimensions of /L5/w5. Given these general considerations, the expression for the probability of any value of &,' is: &,' $ =! %&,( %&,) %! %! (18.22) In other words, we integrate the microstate probability over all of the other 6 1 degrees of freedom, not including &,'. This integral is easy to perform: &,' $ = Λ6 h 6 = Λ6 h 6 P RS T,U V V V T WS T,XWST,YZ S V [,U P E 6 P SV %& G %&,( %&,) %! %! (18.23) In the second line, we factor out the part of the exponential which is due to &,', and the remaining integral can be factored into 3 1 identical onedimensional integrals involving the other momenta. Some additional simplification shows that: &,' $ = Λ6 h 6 = Λ6 h 6 S V [,U P h6 Λ 6 S V [,U P (18.24) Since there is no difference between any of the particles or of the momentum components, we can say that in general the probability that a particle will have a particular momentum at is given by: & ' = 2`Nc d expx & ' 2Nc d y (18.25) This is the MaxwellBoltzmann distribution for one component of momentum for a particle. Equivalent expressions are found for & ( and & ). There are a number of discussion points with regards to this equation: & ' is a probability density, since & ' is continuous. That means that the quantity & ' %& ' gives the differential probability that any particle has one of its components of momentum in the range & ' %& ' 2to & ' + %& ' /2. 8
9 A momentum component & can vary between negative and positive infinity. W] ] Therefore t & ' %& ' = 1. The distribution is Gaussian in momentum. The peak of the distribution lies at & ' = 0, about which it is symmetric. That is, there is no net momentum in either direction. As the temperature is decreased, the peak probability at & ' = 0 increases. This result applies to any system, since we have said nothing about the potential energy function in the system. It also applies to any density, because the final distribution is independent of the system volume. The canonical distribution of atomic momenta given by the Maxwell Boltzmann expression is universal, depending only on the mass of the individual molecules and the temperature of the system. It is independent of the potential energy function and of the density. A plot of this distribution in fictitious units at several temperatures is shown here: P(p) p T = 2.0 T = 1.0 T = 0.7 T = 0.5 Importantly, the & ' distribution can take on two different but equivalent interpretations, & ' gives the probability that, at a random instant in time, a given particle will have an Jcomponent of momentum & '. Alternatively, for a system with a very large number of particles, & ' gives the distribution of & ' over all of the particles at any instant in time. Both interpretations are equivalent at equilibrium. The average kinetic energy of the system can be evaluated using the computed momentum distribution. There are 3 momenta that can contribute kinetic energy. These momenta 9
10 are independent because the kinetic energy is a linear sum of individual momentum component terms, and thus the partition function is factorable in each. We can therefore write: B = 1 2N &,' + &,( + + &,) $ = 3 2N & ' = 3 W] 2N & ' & ' %& ' ] (18.26) In the second line, we simplify the expression since all of the & averages must be the same (there is no difference between particles or components). Substituting in the expression for & ' and simplifying, we find that: B = 3 2 c d (18.27) In plain words, the average kinetic energy is proportional to number of particles and to the temperature. Importantly, we find that At constant temperature for a monatomic system, the average kinetic energy is only a function of the number of particles and the temperature, regardless of the nature of the atoms (their mass or interactions) or the density. We could compute the instantaneous kinetic energy for a number of snapshots of the system, taken randomly in time. We could then find the average kinetic energy. Using the above equation, this would allow us to find an estimate of the temperature. Thus, we say that the average kinetic energy is an estimator of the temperature. What about the distribution of the speed of a molecule, ~? The speed is defined as a positive number, and is the absolute value of the velocity vector for each particle. We can write the speed of a molecule at any point in time as ~ = 1 N & ' + & ( + & ) (18.28) The joint distribution of the three components of a particle s momentum vector is the product of the individual distributions, since the momenta are independent (they are linearly additive in the kinetic energy function). Therefore, 10
11 & ',& (,& ) $ = & ' & ( $ & ) = 2`Nc d 6 expx & ' + & ( + & ) y 2Nc d = 2`Nc d 6 expx! 2Nc d y (18.29) We want to integrate out the orientational component of this distribution to represent the absolute value of the momentum. Letting & ' = & sin cos, & ( = & sin sin, and & ) = & cos : &,, = 2`Nc d 6 expx & 2Nc d y& sin (18.30) Here, the probability distribution acquired a term & sin because of the equivalence of the differential probabilities & ',& (,& ) $%& ' %& ( %& ) = &,, % & % %. Integrating over the angular part, & = 4`2`Nc d 6 & expx & 2Nc d y (18.31) Swapping ~ for &, we arrive finally at the distribution of speeds: 6 N ~ = 4`_ 2`c d a ~ expx N~ 2c d y (18.32) A sample plot with fictitious units at different temperatures is shown below: P(v) v T = 2.0 T = 1.0 T = 0.7 T = 0.5 The average velocity at a given temperature is: ~ = ~ ~%~ = _ 8c d `N a (18.33) 11
12 Thus we see that, at a given temperature, the average velocity we would expect from a particle depends monotonically on the temperature. This relationship, however, only applies to the average velocity. In reality, at constant temperature, the MaxwellBoltzmann distribution shows that particles are distributed in velocity. This does not mean that some particles have lower temperature than others, however, since it is the entire system that has been coupled to the bath. Therefore, The velocity of a molecule is not a measure of its temperature. Instead, the average velocity of all molecules, or of a single molecule over long periods of time, provides an estimator of the temperature. The temperature is not an intrinsic property of a molecule, but an aggregate, statistical property of the bath. The notion of a temperature, therefore, is inherently a property of very large systems of molecules The pressure in the canonical ensemble Earlier we found that, in the canonical ensemble, the average kinetic energy and molecular speed are related to the temperature, and provide an estimator for it. What about the pressure? Is there some way to relate a microscopic average to the pressure? Recall that the Helmholtz free energy is given by o = c d ln. We also know from macroscopic thermodynamics that = %o %. Combining these two equations, we arrive at a relationship between the canonical partition function and the pressure: _% % a, (18.34) Recall that is given by: Substituting this expression: = f,, Λ 6! (18.35) f _%f = c d f % a, % % H #"$ %# %# %# (18.36) Therefore, the pressure only depends on the configurational part of the partition function. It is independent of the kinetic degrees of freedom. Taking the derivative of the configurational integral above requires some careful consideration. Recall that, for the positional degrees of freedom, the associated limits of the integral depend on the system volume. That is, the integral is performed over all 12
13 positions available in a volume. For a cubic box, each particle s xcomponent can vary between 0 and, = 6, etc. Therefore, we need to take the derivative of an integral where the derivative variable is part of the limit. To properly take the derivative, we need to remove 6 from the integral limits. We can do this by a change of variable. We nondimensionalize each particle coordinate by the size of the box, introducing the new coordinates ˆ,' = 6 J, ˆ,( = 6 L, and so on. After we make this substitution, each integral runs between the values 0 and 1, since these are the limits on ˆ. We have: f % % H [ Š "$ %Š %Š %Š (18.37) The term term comes from the fact that we did 3 changes of variables, each of which gave a single 6 factor. Here, we have also rewritten the potential energy function using the new nondimensionalized coordinates. However, since the potential energy depends on the real positions, and not the scaled positions alone, we must scale back to real coordinates: D#,#,,# = D,Š,,Š,,,Š = D 6 Š $ (18.38) The prefactor 6 scales each component of Š. Taking the derivative, we need to use the chain rule because there are two terms involving : f EH [ Š "$ _ %D % ah [ Š "$ G%Š %Š %Š (18.39) Factoring out the common terms and returning the nondimensionalized coordinates to the normal ones, we obtain: f E _%D % agh #"$ %# %# %# = c d 1 f H #"$ %# %# %# 1 f _%D % ah #"$ %# %# %# = c d 1 f _%D % ah #"$ %# %# %# (18.40) To compute the derivative of the potential energy function with respect to volume, we must take the derivative at fixed scaled positions Š : 13
14 %D#,#,,# = % 6 %D Š, 6 Š,, 6 Š $ % = %D %J %ˆ,' %D + + %J %ˆ,' % = I,' $ 6 1 $_ J + I,) $ 6 1 $_ w = 1 3 Œ O # O %w %w 3 3 6a % %ˆ,) %ˆ,) 6a + (18.41) Here we have introduced the force, given by the derivative of the potential energy function with respect to position, Œ O = %D/%# O. The final result shows that the potential energy derivative with respect to volume involves a sum of the force on each particle in a dotproduct with its position. Notice that this sum depends on the particular atomic configuration. Returning to the original expression for the pressure, we obtain: 1 3f R Œ O # O Z H #"$ %# %# %# (18.42) Notice, however, that the integral can be rewritten as a sum over the configurational probabilities by bringing the factor of f inside: Therefore, we can write finally: 1 3 R Œ O # O Z # %# %# %# (18.43) 1 3 Ž Œ O # O (18.44) Here, we are simply taking the average over all configurations according to the canonical probabilities #. That average sum over forces times positions is called the internal virial, and is often denoted by the symbol : Œ O # O (18.45) The expression for the pressure above shows that it relates to an average over forces and positions of particles. If there are no potential energies, and hence no forces, then the virial is zero. In this case, the above expression shows that the pressure simply reduces to the ideal gas pressure. 14
15 18.5 The classical microcanonical partition function For a classical system, the microcanonical partition function requires some special treatment. Recall that one way of writing it for discrete systems is: Ω,, = DOF for DOF for all DOF DOF for (18.46) This equation says that we run through all possible values of the degrees of freedom and count a value of 1 each time we find a microstate that has the same energy as the specified energy. For classical systems, we can t count 1 for each microstate since there are an infinite number of them and Ω would diverge. Instead, when we change the sums to integrals we have to change the Kronocker delta function to a Dirac delta function: Ω,, = 1 h 6! /A!,# 5%! %! %! %# %# %# (18.47) Again, the limits on the integrals are the same as those discussed for the canonical version. Here, we have already added the quantum correction, following the approach for the canonical ensemble. Loosely speaking, the Dirac delta function can be thought of as having the properties: /J5 = J = 0 0 otherwise ] /J5%J ] = 1 (18.48) In simple terms, the Dirac delta function is infinitely peaked around the point where it s argument is zero. In the microcanonical partition function, it selects out those configurations that have the same energy as the specified energy. It can be shown that evaluating the multidimensional integral with the delta function is the same as finding the 6 dimensional area in the space of all of the degrees of freedom where the energy equals specified energy. The microcanonical partition function is not separable into kinetic and potential energy components in the same way that the canonical one is. It is also not separable in a simple way for independent molecules. Moreover, due to difficulties in working with the Dirac delta function, it is almost always much easier to compute rather than Ω for a classical system. Therefore, we will not focus further on the form of Ω in classical systems. 15
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