Lecture 29: Boltzmann Distribution

Key Takeaways

foreign okay so what we're going to do today is we're going to finish our coverage of statistical thermodynamics and um and then we're going to do a couple lectures starting Monday remember tomorrow's uh Friday's a holiday so we'll do a couple lectures uh starting Monday on reacting systems and that includes oxidation processes and that's mainly to half chapter 11 and then um and then that'll be it then we're going to have one more hour uh in on social and personal then we're going to have a game show which I don't know how to run a resume we're going to try it and um and then uh and then uh and then the the last exam which is non-cumulative okay so let's pick up where we left off Max entropy and the boltzmann distribution foreign continued from last lecture so I don't like to do this where I sort of continue with derivation and something across two lectures but um that's how it broke so from last time just um at a high level we decided that we wanted to optimize entropy for an isolated system and we had both an entropy formula so we could do some some math and we could uh write down the S equals zero subject to constraints and the constraints being conservation of energy and conservation of mass and we're going to do this using the method of LaGrange multipliers and we had two of them because we had two constraints Alpha and beta and we ended up with an expression like this some stuff so coefficient independent variables and in order to get our Max entropy condition I should say optimize entropy we had to set these coefficients to zero so there's a very sort of familiar thermodynamic thing that we did and what we determined the first multiplier by normalization so this is um where we left off where we left off was the following we had ni Over N total so this is a distribution function of fractional distributions of particles in state I normalized by the total number of particles and it was like this this multiplier beta as yet undetermined energy of State I normalized by Boltzmann's constant and that whole thing normalized by Q which was the partition function and this was the sum over all possible States e to the beta Epsilon I over Boltzmann's Constant and so this is where we left things so so now what we're going to do is we're going to determine beta we're going to determine beta by considering a microscopic reversible process where some of the particles change their statement so we're going to do is we're going to analyze the s for a process of some State changes I'm going to just go ahead and write down the equations that we've been using and developing for this the S equals log n of I over n total DN of I you can pull that from um you know the last two lectures but now we actually have an expression for this distribution so we're going to use that expression KB sum over I that distribution is e to the beta Epsilon I over boltzmann constant over Q then we have some logs of exponentials and things so we can simplify this a little bit and this equals minus beta sum over I Epsilon I d n of I Plus K beta log of Q sum over I d n of I and taking our definitions this equals minus beta d u Plus k b log of Q d n total so um first thing to note uh right this is zero by construction right this is the LaGrange multiplier uh working right because we we wanted DS to be zero under conditions where D U was zero and DN total were zero and we see that for this distribution it works out so the garage multiplier worked the method worked so I'm going to say 0 by Construction and if if you um I really like this orange pen but life is too short for fading Sharpies okay I I um so by construction this is zero now if you like you can convince yourself that if you have a different distribution function here if you have a different distribution function not this distribution let's say a flat distribution function a distribution function that doesn't depend on energy or has some other functional dependence of energy like I don't know what if some polynomial it could be anything in there you can show that in that case DS um will not be zero yeah so again LaGrange multiplier method works but here's here's the takeaway we want to compare this to what we have uh derived two months ago from the combined statement which is a DS equals 1 over t d u plus p over t DV minus mu over t b m and this is really the important thing here we are comparing the coefficients we're comparing the coefficients and if you remember this is almost like General strategy stuff if you compare coefficients you can buy inspection right set like coefficients to like so what is DS d u at fixed volume and fixed particle number well from our statistical method that's minus beta but from the classical method that's one over temperature and so we can simply conclude that beta equals minus one over temperature so now with that identified we have the Bolton distribution oh there's more paper so with beta equals minus 1 over temperature we have the both distribution number of particles in state I normalized by the total number of particles equals e to the minus E I over k b t normalized by the rotation function or the partition function equals the sum over all states e to the minus E I over k B t and this generally has a a falling exponential form with State energy so State energy e of I occupation n of I right it's a falling exponential function in words this is the distribution that maximizes entropy for an isolated system with and total particles distributed over States with fixed energy levels so unchanging energy levels according to that set of occupation numbers and what is maximum what is the state of Maximum entropy for an isolated system we have a name for that what's the state of Max entropy for an isolated system I think way back to late February even so this is an equilibrium distribution so we have an equilibrium distribution of single particle energy levels or single particle Energies and again um you know you knew this uh already but you know equilibrium does not mean everything is at the lowest energy all semester we've been exploring that consequence so you know if you were if you were at zero Kelvin if you're at zero Kelvin uh or you're a physicist which means you're a frigid heart you're a zero Kelvin this is your distribution function everything's piled up has zero energy right but at any real temperature you you can have you get energetic particles and you get them with a distribution function okay so a couple observations here the first observation is Maxwell distribution sorry it said boltzmann distribution and likelihood foreign likelihood um this is a stealth class on statistics right that's what statistical thermodynamics is it's like a stealth class on statistics so we'll ask how much less likely is it to find a particle in state with energy let's say e let's say e m that's an ugly Epsilon equals e l plus Delta e then in state with energy El F so that's a typical thing that you might need to analyze so let's see NM over NL right that's how you turn the word problem into a math problem how much less likely is it to find this distribution and this distribution or you know within within the equilibrium distribution how much smaller is this number than this number and um you'll find the the partition functions cancel out and you get the following El plus Delta e over K T over e to the minus e l over KT and so the e to the minus e l's cancel and you get e to the minus Delta e over KT okay so this depends depends on Delta Epsilon over KT clearly right it becomes less likely with with increasing energy splitting Delta e and it becomes more likely with temperature right okay so you're going to explore this a little bit uh on the pset and the context of um semiconductors another point the thermal energy right this quantity is really important it shows up in the Natural Sciences over and over and over again and we're starting to see why right the why is what we've just shown right um but the implications are really broad you're going to see this all over the place so let's remind ourselves with both swing constant is it's R Over N A oh that's finally RNA so it's it's the ideal gas constant divided by Avogadro's number we have that from before and this is something uh let's see 1.380e to the minus 23 joules per Kelvin so it has units of entropy um but but that's not a number that I remember here's a number I do remember um and if you work on uh molecular systems or semiconductor systems um pretty much well gee a lot of systems that we work on in dmse this is a useful number to have memorized Bozeman's constant in units of EV for Kelvin 8.617 e minus five why on Earth would you memorize that it sets the energy scale for likely fluctuations right and we have all these words are have meaning right likely fluctuations right those words kind of carry a lot of weight so fluctuation single particle fluctuations with energy on the order of k b t r they're likely it's something you're going to find naturally occurring as opposed to fluctuations which are much much larger than the thermally energy those are unlikely we have all sorts of language for this that comes up in all sorts of contexts for example these fluctuations might be called thermally activated processes right you need a little bit of thermal energy to activate it all right so not only is this a number you should know but the thermal energy at room temperature is a number you should know a 298 Kelvin KBT equals well in joules it's not a number that I know because it's kind of unwieldy but in EV it is a number I know that's the Natural Energy scale of molecular processes so it's 0.0257 electron volts or 25.7 milli electron volts so approximately 25 ml electron volts is the thermal energy at room temperature so if you have a molecular system or an atomic system or an electronic system processes of that energy are likely to be happening spontaneously at ambient temperature processes which much much higher energy are not likely to be happening spontaneously at ambient temperature so you can see how powerful this is for example it's the foundation for the Arrhenius rate equation right the Arrhenius rate equation I'm going to use this the notation from a chemistry class here a rate k equals a pre-factor times e to the minus activation energy divided by Boltzmann's constant times temperature right now this this is um this is a rate percent activation energy right and activation energy and we are going to say hello to 3091 here they may have seen that there so just another context where you're going to see these exponentials and say why where right where'd that come from it comes from the spokesman distribution that we just arrived okay more let's talk about um and for one side only this is a little dense but this is to help you with like with lad let's talk about the Maxwell Voltron distribution and I posted the wiki entry for this on on the on the website um so it's a good it's a good article to read and I think that the lab instructors refer to it as well so we're going to consider particles with energy kinetic energy right kinetic energy is momentum vector modulus squared divided by two times the mass that's kinetic energy of a particle of mass m and um we're in Three space so this is p x squared plus p y squared plus p z squared divided by 2m so that's the energy per particle in state of momentum p okay so what we want to do is we want to figure out the distribution of particles as a function of energy so this distribution distribution of uh distribution in energy I'll use f of B for consistency with Wikipedia it depends on both the Maxwell sorry the boltzmann distribution I had an error in my nuts there the Boston distribution n of I and for n total equals Q to the minus 1. exponent and it's going to be minus momentum of State I modulus squared over two m k t right so it depends on that they're both in distribution and also it depends on the density of states in momentum this concept of density of states is not something that we've covered in this class if you are in um O2 o29 you've discussed this you're going to see it in the in the in the lab but this really is a preview of what comes um in the fall when density of States becomes kind of uh um will be second nature for you in the context of quantum mechanics but density of States means well this is the bolstered distribution for a given State I with this energy how many such states are there right how many such states are there um so 3D differential four equals four pi um P quantity squared DP to get the 4 Pi there because I integrated the spherical coordinates so four Pi stradians all around the sphere and that equals 4 Pi m root 2 m e d e so this is a distribution of states in energy e and you combine these two things you combine the density of states and the bolts and distribution and you get V Maxwell volts in distribution distribution of energetic particles zipping around on the kinetic energy included as a function of energy and it's two e over pi 1 over KT to the three halves e to the minus E over k t so this is uh this is the the Maxwell boltzmann distribution and if you plot for example speed particle speed over probability right here's a low temperature you have something that's pretty sharply spiked here the intermediate temperature and then at high temperature this is a series of increasing temp okay and so this has this has the feature that um there are more particles that that go faster at higher temperature right you can see that coming from from this exponential and there's also a vanishingly small number of particles at zero speed right and there's going to be some mean speed that characterizes the distribution and that mean moves with temperature okay so I haven't derived this in detail because you know we haven't spent time on density of States but we're just exploring implications of that of that Baltimore distribution here's another implication okay um the last thing I want to cover which with only a couple slides to go is the concept of ensembles which is really just um 3030 Readiness right we're doing this just to show you that you know some Concepts which you're not going to use in this class but um you will use in in kinetics and macrostructure in the fall so before I move on to discuss ensembles I want to pause and take questions on the boltzmann distribution or Maxwell boltzmann or anything else Arenas or anything else related who's seen the boltzmann distribution or this okay who's seeing the Arrhenius rate law there's not a lot of participation on zoom and so I understand that um I'm guessing that a large number of people raise their hands um and then if I asked you about the Bolton distribution to be a small number and if I ask you about the Maxwell bullets and distribution it'd probably be an even smaller number but I bet that everyone's seen something of this form somewhere before so I hope it's a little bit satisfying for you to see where it comes from right and to understand the principles behind it maximum entropy okay so let's talk about ensembles ensembles are what I look up a different definition of Cambridge Dictionary it says a group of things are people acting together or taken as a whole so this is um you know a concept and and we're going to use it a lot in statistical thermodynamics if not necessarily in this class but you know like I said this is 3030 Readiness here all right if I didn't tell you about this again trouble with Professor who all right so we're going to start with the micro canonical Ensemble and this is something that you've seen before micro canonical something called the micro canonical ensemble this is the set of all possible microstates so the type of stuff that we've been doing instead of all possible microstates of a system with fixed energy volume and particle number right so a system with fixed energy volume and particle number that is um that's an isolated system right for an isolated system equilibrium is state of Max entropy that's what we were just analyzing all right so this thing which we've been doing for the last two and a half lectures this is called the micro canonical Ensemble just gave a name to it and so the partition function that we've been analyzing is actually better known as a micro canonical partition function normally written as Q as we've been doing and it is a sum over single particle energy states some Over All possible single particle States and we know that we're at equilibrium equilibrium with s maximized for what for the Boltzmann's distribution so the probability of finding particle on state I equals number of particles that say I at that distribution over the total number of particles equals e to the minus e i you're getting sick of seeing me write this by now right prob of finding a single particle in state I with energy e by okay so um this is a repeat of everything we just did I just give it a new name I called it micro canonical so again that name's kind of kind of highfalutin but here you have it but we can generalize so now I'm going to tell you about a different type of Ensemble so we've been analyzing the micro canonical now I'm simply going to tell you about the existence of a canonical ensemble and please don't ask me where these names come from a canonical ensemble is the set of all possible microstates of a system with fixed volume and n but not energy so now we're allowing the system to exchange energy the system is closed it's rigid and closed so volume and mass are conserved system is rigid and closed but can Exchange energy and form of heat with surroundings and the system energy which we're now going to give a label to U sub Nu the system energy can fluctuate so I want to draw for you a representation of the micro of the canonical Ensemble so we're going to imagine um grab a defined tip marker we're going to imagine system labeled I with energy U sub I but this thing is um in thermal contact with a lot of other systems so we're going to imagine a whole population or if you like an ensemble of systems all with fixed volume and fixed particle number but with diathermal walls so if heat is like sound you have to imagine this is an apartment building that's really really noisy you hear all your neighbors so the volume the volume mistakes and the particle number is fixed but you're exchanging energy right through the walls but this Ensemble as a whole so I've drawn an ensemble right I've drawn a set or a group considered as a whole right I've drawn a bunch of subsystems now that can exchange energy with each other they each have the same volume well not as I've drawn it but you can imagine they each have the same volume each have the same particle number but the insoluble as a whole is isolated so all these ensembles you have to imagine this being a very very large number here not just four by six or whatever I've drawn um is isolated and so this is an ensemble and this is an ensemble has its own partition function it's known as the canonical protection function and that's typically written as Z and this is now a sum as before a sum Over States but we're not summoning over single particle States we're summing over system energy states some over all possible States of the system so it's distinct from before we were summing over single particle States and equilibrium hold on when we have equilibrium in an isolated system it was Max entropy now I think back two months ago when we had equilibrium for fixed temperature and fixed pressure that was minimum Gibbs but there was another case which we haven't spent a lot of time on it was equilibrium at fixed volume and fixed temperature do you remember what thermodynamic potential we use when we have a system at fixed volume and fixed temperature is it Heim Holtz equilibrium at minimum f again we haven't used that this course basically is minimized for the distribution probability of finding system in state new equals e to the minus U Nu over KT over Z so it has the same functional form as the Bolton distribution except instead of single particle energy this is energy of the system as a whole and instead of maximizing entropy it minimizes how much free energy so there's something very similar that's happening here similar to when we started with entropy and then we moved to helmhot's energy and then we move through Gibbs energy as we change the boundary conditions of the system and I think you'll remember in problem set two we really dove into this so there's something very similar that happens in statistical thermodynamics where you start with the micro canonical Ensemble and its partition function and then you go to the canonical Ensemble and its partition function and there's more there are other ensembles the next one is called the Grand canonical Ensemble and that corresponds to situations with fixed pressure and temperature We're Not Gonna bother to introduce that so this is the boltzmann distribution but it's for the canonical ensemble in that so in this case what was it that you thought you held constant um so this is equilibrium at fixed temperature right when you have when you have um an ensemble of systems that can't exchange volume and they can't exchange particle number but they can exchange heat energy they reach equilibrium when the temperature becomes constant right that's thermal equilibrium and so that's what we've done here we've set up a system a collection an ensemble of subsystems that reach thermody thermal equilibrium with each other because they can exchange heat energy it all works out you end up getting minimum f for this distribution this is the probability probability of finding system in state new with energy you have new that's what that probability is okay so I want to leave you with some final words on partition functions we've seen some partition functions Q and z and and there's there are others which we won't cover in this class right we haven't really covered these and this won't be on the exam I mean there's just sort of commenting on what comes next again this is this is in service to later classes in our curriculum so the partition functions they describe they describe they describe equilibrium and fluctuations the partition functions really describe the entire thermodynamic properties of the system um so for instance they are the basis for calculating thermodynamic properties and expectation values that is most likely quantity so for example for example the mean energy of a system can be shown to be this derivative d log Z 1 over KT so that's a little bit formal can't necessarily calculate anything yet but in principle if you know the partition function you can calculate the mean energy the heat capacity of a system is 1 over KT squared D Squared log z d 1 over k t squared the entropy of a system can be shown to be mean energy divided by temperature plus k b log Z the helmo three energy of a system is minus k b t log Z and it's not just equilibrium properties it's properties of fluctuations so this gets really fun uh in a later class and has a lot of implications so for example the root mean squared energy fluctuation of a system is another quantity that can be calculated from the partition functions so you know I'm just showing you things but you know if you have the partition function for a system you can calculate the entire thermodynamic properties of that system so another example is in in the textbook if you look into Hoff uh they calculate the partition function for a couple different systems a two level system and then a system of um a monoatomic gas and then by applying formulas such as these they derive the ideal gas law so it's sort of the ideal gas law from first principles right it's no longer empirical so I'm going to leave you with some thoughts um at this point often students say ah this is the real stuff this is somehow the most fundamental knowledge of thermodynamics is a partition functions right so why did we mess around with all that classical thermodynamics right if the partition functions are the keys to the kingdom right why not just start there and the answer is that although this this approach is beautiful and elegant and very powerful in instances it also is extremely limited because we don't know and we can't calculate the partition functions for all but the most toy problems and you can see that we if you take thermodynamics in a physics department you sort of start from this approach you start from the statistical thermodynamics approach because um physicists Love Story problems right things that's intellectually rich but you know may or may not apply to the real world we're a little bit more applied to give you an example there's a there's a toy model of magnetism known as the ising model and the ising model says that you can have um spins on a lattice or magnetic moments I should say on the lattice and they can face up or down that's it so they have a one-dimensional chain and each spin can face up or down and the energy of the system depends on on the orientation of the nearest neighbor it's a very simple model of a magnet and so if you if you take a course on magnetism or you take a thermodynamics course in the physics department you'll probably calculate the partition function for a 1D eising chain it's not that hard it's kind of fun all right however if we simply make it a two-dimensionalizing system and I'm drawing the direction of the arrows at random here just sort of showing you right so we just made this from one dimensional which is definitely an imaginary thing to a two-dimensional which is still imaginary but we're getting there right we're at least getting close to real space calculating a partition function for this model um is one of the grand intellectual achievements of 20th century physics and one large answer in Nobel Prize it's horrendously complicated mathematically um I've never been able to follow even the first page of that it's not in any textbooks it's too long um so let's say I want to model a real magnet in three dimensions forget it I mean unless you're kind of Nobel Prize quality um that's that's an open problem you have at it but basically the point is that for real systems that we care about as Engineers this sort of is a magical construct in theory this exists in theory it allows you to calculate all the properties of the system from first principles but in practice it just Remains the intellectual exercise in practice we have to if you like go with a much more powerful which is the root of classical thermodynamics with its postulates and and observations and data and models