Topics in Modern Statistical Physics
In the fall semester of 2023 I will teach the course Topics in Modern Statistical Physics at the University of Warsaw. This year the focus will be on the application of statistical mechanics to classical systems, such as simple (atomic liquids) and complex fluids (soft matter, biological systems). To get a complete view on the uses of statistical physics, we recommend also the students to follow a course on quantum many-body theory (which will not be discussed in this year's lectures).
Lecture notes are available here and the exercises can be found here. Slides with illustrations can be found here. All documents will be continuously updated throughout the course. For comments on both documents, please contact me directly.
Schedule: The course will start on 4 October 2023.
Lectures: Every Wednesday, 9.15-12.00, room 2.06, Physics building.
Exercise classes: Every Thursday, 16.15-19.00, room 1.02, Physics building.
Examination:
Mid-term exam (kolokwium) on Wednesday 06.12.2023 at 09:00-12:00 in Room 2.06 (instead of lecture).
Final exam on 07.02.2024 at 09:00-12:00 in Room 1.38.
Retake exam (oral) on 20.02.2024 for eligible and interested students in my office, room 5.32.
Check here for latest updates on the time schedule. Both exams are closed book.
Prerequisites: The students entering the course are expected to have completed an introductory lecture of statistical physics and thermodynamics as well as the standard undergraduate level courses (classical and quantum mechanics, electrodynamics). Some knowledge on fluid mechanics and stochastic processes is useful.
Assessment: Mid-term exam (30%), hand-in exercises (30%), final exam (40%). Students can earn a bonus point (max. 0.5) to the final grade by giving a presentation that fits within this lecture series. The retake exam is an oral exam, and can only be taken for students who failed the final exam (less than 50% of the points) and have obtained at least half of the total earnable points of the hand-ins.
Material discussed in lectures (L=Lecture) In grey, the tentative plan is shown.
L1 (04-10-2023): Overview of the course, examples of classical many-body systems (simple fluids, complex fluids), introduction to soft matter with examples (colloids, liquid crystals, polymers, granular matter) and its properties, such as viscoelasticity. The lecture followed roughly the first two chapters of the book Soft Condensed Matter by R.A.L. Jones and Chapter 1 of the lecture notes.
L2 (11-10-2023): We discussed virial expansions, and showed how to derive terms in the virial expansion using diagrammatic methods or via direct computation. We argued that the virial expansion -although useful for dilute systems - is slowly converging for dense fluids. Furthermore, we have taken a brief look at hard-sphere freezing. See Sec. 2.3 from the lecture notes.
L3 (18-10-2023): In the lecture, we discussed the importance of density-density correlation functions, defined as the ensemble average of suitable combinations of the (classical) density operator. We focused on some quantities derived from it, such as the radial distribution function g(r). We interpreted g(r) as a conditional probability and in terms of a potential of mean force. Finally, we derived how to use g(r) to obtain the complete thermodynamics of the system using the caloric, virial, and compressibility route. See Secs. 2.4 and 2.5 from the lecture notes.
L4 (25-10-2023): Today we discussed how the static structure factor can be determined using scattering measurements using X-rays or neutrons. We introduced the Ornstein-Zernike (OZ) equation which defined a new type of correlation function called the direct correlation function, which asymptotically goes to minus the pair-interaction potential. In order to solve the OZ equation one has to provide a closure relation, however, there are only approximate closures available. We discussed the random-phase approximation, mean-spherical approximation, and the Percus-Yevick approximation (the latter being an excellent approximation for hard spheres). See Secs. 2.5 and 2.6 from the lecture notes. We ended with discussing critical opalescence using the Ornstein-Zernike equation, see also the end of Sec 5.6 in the book "Theory of simple liquids" by J-P. Hansen and I.R. McDonald.
No lecture on Wednesday 01-11-2023.
L5 (08-11-2023): We introduced the language of functionals, and we have shown some examples of the functional derivative and some properties derived from it, such as the functional Taylor expansion and the functional chain rule. Using functionals, we introduced density functional theory, a useful framework to study inhomogeneous fluids. We proved that there exists a unique functional of the density which holds for every external potential. This so-called intrinsic Helmholtz free energy functional equals the Helmholtz free in the absence of an external potential. Moreover, we have proven an important variational principle which determines the equilibrium density profile. When applied, we recover the condition that the chemical potential in the grand-canonical ensemble should be constant throughout the fluid. See Sec. 3.1 in the lecture notes. See also Sec. 3.2 in the book of Hansen and McDonald for an introduction to functionals.
L6 (15-11-2023): We have shown that DFT in a natural manner generates two hierarchies of correlation functions: (i) the density-density correlation functions that are obtained via functional differentiation of the grand potential viewed as a functional of the external potential with respect to the intrinsic chemical potential, (ii) direct correlation functions via functional differential of the intrinsic Helmholtz free energy functional with respect to the equilibrium density profile. Having two generating functionals that are related via Legendre transform, led to the emergence of the inhomogeneous Ornstein-Zernike equation. We have derived formal exact expressions for the excess Helmholtz functional by integrating with respect to the particle density and interaction potential, which formed a basis for approximative functionals. Furthermore, via the Percus' test particle formulation, it allowed us to derive approximative closure relations for homogeneous bulk systems, such as the Percus-Yevick approximation. Finally, we started with discussing one-dimensional hard-rod systems for which the Helmholtz functional is analytically known. See Secs. 3.2-3.6 in the lecture notes.
L7 (22-11-2023): We discussed the idea behind fundamental measure theory. Inspired by the exact result of hard rods in one spatial dimension, we determined the excess functional by imposing that (i) the functional must be a local functional of weighted densities, and (ii) the exact low-density limit must be retrieved. We showed how a deconvolution of the Mayer function in 3D helped us to construct the necessary weight functions in 3D from which the excess functional can be determined. We discussed improvements such as the White Bear functional that uses the Mansoori-Carnahan-Starling-Leland equation of state as an input. We discussed also shortcomings of the FMT like functionals such as not being able to predict hard-sphere freezing and the concept of dimensional crossover. Finally, we derived the van der Waals equation of state and discussed that attractions (already on the mean-field level) give rise to a thermodynamic instability. We discussed the spinodal (the line which indicates the limit of thermodynamic stability) and the binodal (the phase boundaries). The latter can be obtained via a common-tangent construction, and we discussed shortly how the metastable region and absolutely unstable region differ in kinetics. See Secs. 3.6 and 3.7 from the lecture notes on FMT. For van der Waals gas, see Sec. 4.3 in the book "Basic concepts for simple and complex liquids" by Jean-Louis Barrat and Jean-Pierre Hansen. Some additional background on phase transitions can be found in Chapter 3 of the book "Soft Condensed Matter" by Richard Jones.
L8 (29-11-2023): In this lecture we have taken a deeper look at the gas-liquid phase transition and, in particular, to the gas-liquid interface. We did this by using density functional theory within the square-gradient approximation. We discussed in detail how one can attribute a microscopic picture for the coefficients appearing in this approximation. When the square-gradient coefficient is constant in density, we retrieve the van der Waals theory for the gas-liquid interface. Solving for the Euler-Lagrange equations with suitable boundary conditions gives rise to the density profile of such an interface. We used this profile to define the surface tension as being the excess grand potential per unit area over the bulk system. We discussed briefly the effects of surface tension, such as Laplace pressure and the fact that a curved surface enhances solubility in liquid-liquid mixtures. We used this argument to qualitatively describe Ostwald ripening: the growth of big droplets at the expense of smaller droplets. At the end of the lecture we introduced the concept of wetting via the Young's law.
L9 (07-12-2023): We discussed the mid-term exam.
L10 (13-12-2023): Starting from a short discussion of linear irreversible thermodynamics, we derived a generalised diffusion equation where the driving force is the intrinsic chemical potential. Using for this equation the square-gradient functional, we derived the so-called Cahn-Hilliard equation which is a generalised diffusion equation that qualitatively describes spinodal decomposition. We discussed the linearised solution to this equation and also qualitatively how non-linear effects changes our findings from the linear analysis. We proceeded by discussing nucleation and growth and how this effect differs from spinodal decomposition. Finally, we considered surface phase transitions, related to wetting (drying) of a bulk fluid in contact with an attractive (repulsive) wall. For spinodal decomposition and nucleation and growth, see Chapter 3 of the book "Soft Condensed Matter" by Richard Jones. In Chapter 9 of "Basic concepts for simple and complex liquids" by Jean-Louis Barrat and Jean-Pierre Hansen, there is an alternative exposition of spinodal decomposition and 10.6 for nucleation and growth. For wetting transitions, see, for example, section IIIB of the review paper by Nobel Prize winner Pierre-Gilles de Gennes, which can be found here.
L11 (20-12-2023): We discussed the concept of an effective interaction potential that emerges when one partially integrates out degrees of freedom in the partition function. By keeping the degrees of freedom that we are interested in, we obtain a conditional free energy that acts like a potential of mean force. We gave various examples of this concept, such as electrostatic interactions in a dielectric medium, the angular-averaged dipole-dipole interaction (Keesom interactions), the van der Waals interaction, and depletion interactions. The latter is obtained by integrating out degrees of freedom of an "uninteresting" species in a multi-component system. We have shown that the depletion attraction is driven by entropy, or alternatively by osmotic pressure. See Sec. 4.3 in the book of Richard Jones "Soft Condensed Matter" and Sec. 2.7 in "Basic concepts for simple and complex liquids" by Jean-Louis Barrat and Jean-Pierre Hansen.
No lectures on Wednesday 27-12-2023 and 03-01-2024 (Christmas holidays).
L12 (10-01-2024): We discussed the statistical physics of ionic solutions in bulk within the Debye-Hückel approximation. In particular, we have derived the limiting law which shows the exact lowest order correction to the ideal gas free energy. We have discussed the possibility of a gas-liquid transition in such a system and why it is in practice hard to observe. Subsequently, we discussions ions in the presence of external charged surfaces (walls, particles) within the framework of density functional theory. Employing a mean-field approximation for the electrostatic interactions, and an ideal-gas functional for the non-Coulombic part, we have derived the so-called Poisson-Boltzmann equation. The concept of electrostatic screening has been introduced. In the final part of the lecture, we discussed effective interactions in ionic solutions, such as the disjoining pressure between two flat walls and the DLVO potential between charged spheres. The applicatibility of the latter has been demonstrated in the context of charge stabilisation of charged colloidal suspensions. Further reading on this topic: Sec 4.3.3 of the book by R.A.L. Jones and Secs. 7.6 and 7.7 of "Basic concepts for simple and complex liquids" by Jean-Louis Barrat and Jean-Pierre Hansen. One could also have a look at Secs. 10.6 and 10.7 in the book of Hansen and McDonald for a more advanced discussion.
L13 (17-01-2024): In this lecture we discussed the Landau theory of phase transitions. We started with the Ising model with nearest-neighbour actions and discussed on the basis of the mean-field theory how to write the free energy in terms of an order parameter. Subsequently, we showed that a general way of introducing the order parameter without neglecting correlations can be formally performed using the Hubbard-Stratonovich transformation. We discussed the concept of spontaneous symmetry breaking, and highlighted the difference between breaking a discrete symmetry of the underlying Hamiltonian and a continuous symmetry. The latter we demonstrated using the classical Heisenberg model. Finally, we introduced the concept of a nematic liquid crystal, a phase of matter that is similar to a magnet with the addition of the so-called up-down symmetry. We discussed the corresponding expansion in terms of the tensorial order parameter, which is called Landau-de Gennes theory. See Chapter 5 of the book of David Chandler, Introduction to modern Statistical mechanics for discussion of the mean-field theory of the Ising model. For Landau theory of phase transitions, one could consult the book "Ultracold quantum fields" by Henk Stoof, Dennis Dickerscheid and Koos Gubbels. Specifically, the Hubbard-Stratonovich transformation applied to the Ising model can be found in the book "Field Theory, the renormalisation group, and critical phenomena" by Daniel Amit in Section 2.5.
L14 (24-01-2024): In this lecture we discussed the physics of liquid crystals in more detail. We discussed the types of liquid crystals (thermotropics vs lyotropics) and the various liquid-crystalline phases (nematic, smectic,...). We discussed lyotropics in the context of hard rods within the Onsager model. We showed that a isotropic-nematic phase transition can occur purely based on repulsions, contrasting thermotropics for which the transition occurs because of attraction. Furthermore, we discussed some characteristic properties of liquid crystals, such as nematic elasticity, anchoring, and the response to external fields. Briefly, we have mentioned the concept of a topological defect, and that nematic elasticity causes the core of a defect to be isotropic. We finished the lecture with some applications, such as the Frederiks transition, and explained how this effect can be used to create a liquid-crystal display. For further reading one could have a look at Chapter 7 of the book by Richard Jones. A more advanced treatment can be found in Chapters 2 and 3 in "Physics of Liquid Crystals" by de Gennes and Prost.
Material discussed in tutorials (T=Tutorial)
T1 (05-10-2023): Quick reminder on statistical mechanics and the classical limit from the quantum partition function, see Sections 2.1 and 2.2 of the lecture notes. For pedagogical reasons, we performed the classical limit only for the one-particle partition function, see the lecture notes for the many-body case. We worked on Exercise 2.1 as an example of an exactly solvable partition function of classical interacting particles.
T2 (12-10-2023): Exercises 2.2, 2.3, and 2.4a,b. These exercises are meant to practice with virial coefficients. As a hand-in we required either Problem 2.2 or 2.3 and 2.4a,b.
T3 (19-10-2023): Problem 2.4c, 2.5. In these problems, we derive the virial expansion of g(r), and have a closer look at the virial route to thermodynamics.
T4 (26-10-2023): Problem 2.6 and 2.7. We will practice with the Ornstein-Zernike equation within the random-phase approximation and the Percus-Yevick approximation. As a hand-in we required Problem 2.7.
No tutorial on 02-11-2023.
T5 (09-11-2023): Problem 2.8 and 3.1. We derive and apply thermodynamic perturbation theory and study some general properties of the density functional. Lastly, we will apply DFT to an ideal gas in external field.
T6 (16-11-2023): Problem 3.2 and 3.3. We will practice with the local-density approximation and density expansions. Both were a hand-in exercise.
T7 (23-11-2023): Problem 3.6. We will practice with one-dimensional hard rods in an external potential, represented in terms of weighted densities. Problem 3.6 was a hand-in.
T8 (30-11-2023): Problem sheet week 8. We will discuss in detail the square-gradient approximation and apply it to the van der Waals model for the gas-liquid interface. Furthermore, we will have a look at the law of corresponding states, which introduces one of the first notions of universality in the phase behaviour of substances.
T9 (04-12-2023): Question hour before the midterm exam.
T10 (14-12-2023): Problem sheet week 10. As a hand-in we required one of the problems of this document.
T11 (21-12-2023): Problem sheet week 11.
No tutorials on Thursday 28-12-2023 and 04-01-2024 (Christmas holidays).
T12 (11-01-2024): Problem sheet week 12. As a hand-in we required one of the problems of this document.
T13 (18-01-2024): Problem sheet week 13. As a hand-in we required one of the problems of this document. In this tutorial we will have two student presentations (each 20 min+10 min discussion/feedback).
T14 (25-01-2024): No new exercises. In this tutorial, we will have five student presentations (each 20 min+10 min discussion/feedback).