A Deeper Look into Phase Space: The Liouville and Boltzmann Equations Brian Petko Statistical Thermodynamics Colorado School of Mines December 7, 2009
INTRODUCTION Phase Space, in a sense, can be a useful crutch for understanding statistical mechanics. It represents a picture that describes the problem we are trying to solve. Because I am a very visual person I try to relate many problems to a picture in phase space and than justify certain trends or concepts from this. It is therefore the first place I turned as I came to a problem in my research. This paper will focus on a more in depth look at phase space through the Liouville and Boltzmann equations and their significant meanings. It will also show how a deeper understanding of phase space has proved that I am asking the wrong questions with my research problem and the direction I should be taking. BACKGROUND My research consists of modeling lithium ions in polymer electrolytes for the purpose of determining a transport mechanism of lithium ions. This may be useful for the future design of lithium- ion polymer batteries with regards to the type and length of polymer chains and/or type of anion used. After building smaller simulations to gather a better understanding of how the program worked I was ready to build a large scale molecular dynamics simulation. The simulation consisted of 30 Polyethylene Oxide chains with 26 repeat units, 50 Lithium Ions, and 50 anions. This comes out to be 2500 atoms. The simulation is run for 100 picoseconds with
each time step being 2 femtoseconds. These parameters were based off of previous works. The problem occurs with the idea of what I originally thought of as equilibrium but now know to be ergodicity. As others in the class may be interested in molecular simulations I thought it may be relevant to bring up the program Packmol and its capabilities while describing where my problem stems from. Packmol creates an initial point for molecular dynamics simulations by packing molecules in defined regions of space. The packing guarantees that short range repulsive interactions do not disrupt the simulations. This proved to be very useful as writing a coordinate file for 2500 atoms with the majority of atoms being constrained by being in the same chain very difficult. I simply made a coordinate file for one polymer chain, one lithium atom, and one anion. Packmol takes this file and places the number of molecules of each that you specify into a box of size that you specify. This allows you to get the density you want with a fairly random starting configuration. It will also keep a certain distance of your choosing between atoms as it packs it into the box. This will help ensure that your simulation will not blow up due to inter- atomic forces. During my first attempt at building the systems I constructed a linear PEO chain, length of 80 angstroms, as an input to Packmol. My box size was 50 angstroms. As you may expect, the only way to fit this chain in the box is to have the chain start in one corner and end in the cattycorner. This was unacceptable as it creates an artificially high density in the center of the simulation. My solution was to redo the PEO coordinate file to have a fold in it, making its length approximately 35 angstroms. The resulting Packmol solution is shown in Figure 1.
Figure 1 As can be seen in Figure 1 all the polymer chains have a bias to have little entanglement which continues throughout the course of the simulation. In order to calculate properties, however, you would want something that is a good representation of what is found in Lithium polymer batteries. This would include entangled polymer chains. Therefore, I concluded that the simulation was not at equilibrium, it still had memory of its starting configuration. The previous works on this topic do not discuss any requirements necessary for the system to reach equilibrium. I therefore set forth to derive an equation that would allow me to determine when my system reaches equilibrium.
I thought it would be a good idea to start with a simpler system so I was pointed in a direction towards phase space and the kinetic theory of gases. The following is a review of the Liouville and Boltzmann equations, their meaning in phase space, and how phase space has redirected my approach. THE LIOUVILLE EQUATION The Liouville equation is simply a phase point balance on phase space. Let f N be the phase space distribution function. The density is normalized by the following function: If we select a volume (v) of phase space than the number of points found in v at any given time is given by: We than take the derivative of each side with respect to time. Next, we set up a microstate balance on the volume v. Since phase space points can neither be created nor destroyed the change in points found in v is simply In Out. This is the rate at which phase points flow through the surface containing v. By using Gauss theorem we transform it to a volume integral to get:
By setting this equal to the general equation from before we get: This looks like a continuity equation. By substituting in for the second term using its definition and the fact that the second term in the definition is 0, and by substituting in for the Hamiltonian equation we get the Liouville equation: In Cartesian coordinates the Liouville equation is given by: This can be written in a way similar to the Schrodinger equation by introducing the Liouville operator.
THE BOLTZMANN EQUATION The Boltzmann Equation begins with a dilute gas where only two- body interactions are considered. We start with an equation for phase space density of species j when there are no collisions. Phase space density will than follow the equations of motion as seen by: However, because of collisions, some of the species enter the non collision stream while others leave it. We will let gamma positive and gamma negative represent these quantities. On a differential piece of phase space we get: By subtracting f j from the right hand side, dividing each side by dt and expanding the left hand side it can be shown that: This looks very similar to the Liouville equation. To find an explicit expression for the gamma functions let s consider a molecule of type j at r with a certain velocity. The probability that j will collide with another species i over the time interval dt at an angle to make species j either leave a stream is given by:
This assumes that the velocities and positions are uncorrelated. By making the same argument for species j entering a stream we arrive at the Boltzmann equation: By manipulating the Boltzmann equation we arrive at several interesting conclusions. First, multiply the Boltzmann equation by some property psi and integrate over v: It can be shown that the right hand side is given by: When psi equals mass, momentum, or energy it is obvious that the right hand side goes to 0. The left hand side goes to the corresponding continuity equation. A very interesting aspect of the Boltzmann equation is found through what is called the H- theorem: We recognize this as something similar to the Gibbs entropy formula. If we differentiate with respect to time we get:
Due to the conservation of phase points the second term goes away. By substituting in the Boltzmann equation we get: The first two terms go to zero as f goes to zero at the walls and as v goes to plus or minus infinity. This may be simplified to: This function is always less than or equal to zero, giving: Since H is related to the negative of energy, this equation satisfies the second law. CONCLUSION This idea that the H function must decrease was met with a lot of criticism. Classical equations have no preferred direction in time. Two atoms that collide and separate do not care if time is moving forward or backwards. The classical equations give the same equation. The Boltzmann equation can be thought of as a classical equation however the H- theorem shows that the Boltzmann equation has a direction in time, but how can this be? Boltzmann argued that the gamma functions described before are not the number of collisions but the probability of a collision. Therefore, the H- Theorem does not necessarily decrease but that the probability of it decreasing is far greater than it increasing. Another
argument was that this violated the Poincare recurrence time which states that a finite system will return arbitrarily close to its original state. Boltzmann argued that this recurrence time is sufficiently large enough for relative systems that it can be ignored. If we look at the H- function as having a probability of increasing than the system may return arbitrarily close to its original position but because it is extremely unlikely we would have to wait a very long time for this to happen, maybe on the order of 10 to the trillion years. These ideas lead to the following picture of phase space and the following conclusions to be research problem. An overwhelming majority of phase space is the system at equilibrium. Even if we find the system in a non- equilibrium region it appears to be going to equilibrium. Rarely will it pass back through a non- equilibrium region (due to fluctuations). This is true for both the forward and backward time trajectories. Figure 2 is a visual representation of phase space.
Figure 2 The black circles may represent a region in which a phase space point may become trapped. The example we used in class was a 2D box of particles with perfectly smooth walls and molecules traveling parallel to each other and perpendicular to the wall. What I did not understand before is that my system reaches equilibrium very quickly. There is a possibility that in a polymer melt all the polymers are folded once and relatively untangled but that does not mean that this configuration is not an equilibrium configuration. However, it is very unlikely to find in experiment and therefore a poor representation of what I am trying to model. I would therefore like to get into a different region of phase space equilibrium that is more likely found in an experimental melt; however the recurrence time for polymers is very large compared to something like a liquid or gas. In order to counter this I have decided to heat the simulation up drastically so that the system is more
like a liquid or gas. I will than cool it back down to take measures of properties. I will than heat it back up followed by cooling many times to allow myself to travel to many equilibrium regions of phase space in a shorter period of time. The next step is to determine how hot and how many times to heat and cool in order to have relatively ergodic data.