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Lecture Notes for Statistical Mechanics of Soft Matter

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Lecture Notes for Statistical Mechanics of Soft Matter Lecture Notes for Statistical Mechanics of Soft Matter 1 Introduction The purpose of these lectures is to provide important statistical mechanics back- ground, especially for soft condensed matter physics. Of course, this begs the question of what exactly is important and what is less so? This is inevitably a subjective matter, and the choices made here are somewhat personal, and with an eye toward the Amsterdam (VU, UvA, AMOLF) context. These notes have taken shape over the past approximately ten years, beginning with a joint course developed by Daan Frenkel (then, at AMOLF/UvA) and Fred MacK- intosh (VU), as part of the combined UvA-VU masters program. Thus, these lecture notes have evolved over time, and are likely to continue to do so. In preparing these lectures, we chose to focus on both the essential techniques of equilibrium statistical physics, as well as a few special topics in classical sta- tistical physics that are broadly useful, but which all too frequently fall outside the scope of most elementary courses on the subject, i.e., what you are likely to have seen in earlier courses. Most of the topics covered in these lectures can be found in standard texts, such as the book by Chandler [1]. Another very useful book is that by Barrat and Hansen [2]. Finally, a very useful, though more advanced reference is the book by Chaikin and Lubensky [3]. Thermodynamics and Statistical Mechanics The topic of this course is statistical mechanics and its applications, primarily to condensed matter physics. This is a wide field of study that focuses on the properties of many-particle systems. However, not all properties of many- body systems are accessible to experiment. To give a specific example: in an experiment, we could not possibly measure the instantaneous positions and velocities of all molecules in a liquid. Such information could be obtained in a computer simulation. But theoretical knowledge that cannot possibly be tested experimentally, is fairly sterile. Most experiments measure averaged properties. Averaged over a large number of particles and, usually, also averaged over the time of the measurement. If we are going to use a physical theory to describe the properties of condensed-matter systems, we must know what kind of averages we should aim to compute. In order to explain this, we need the language of thermodynamics and statistical mechanics. 1-1 In thermodynamics, we encounter concepts such as energy, entropy and tem- perature. I have no doubt that the reader has, at one time or another, been introduced to all these concepts. Thermodynamics is often perceived as very abstract and it is easy to forget that it is firmly rooted in experiment. I shall therefore briefly sketch how “derive” thermodynamics from a small number of experimental observations. Statistical mechanics provides a mechanical interpretation (classical or quan- tum) of rather abstract thermodynamic quantities such as entropy. Again, the reader is probably familiar with the basis of statistical mechanics. But, for the purpose of the course, I shall re-derive some of the essential results. The aim of this derivation is to show that there is nothing mysterious about concepts such as phase space, temperature and entropy and many of the other statistical mechanical objects that will appear time and again in the remainder of this course. 1.1 Thermodynamics Thermodynamics is a remarkable discipline. It provides us with relations be- tween measurable quantities. Relations such as dU = q + w (1.1) or dU = T dS P dV + µdN (1.2) − or ∂S ∂P = (1.3) ∂V ∂T T V These relations are valid for any substance. But - precisely for this reason - thermodynamics contains no information whatsoever about the underlying microscopic structure of a substance. Thermodynamics preceded statistical me- chanics and quantum mechanics, yet not a single thermodynamic relation had to be modified in view of these later developments. The reason is simple: ther- modynamics is a phenomenological science. It is based on the properties of matter as we observe it, not upon any theoretical ideas that we may have about matter. Thermodynamics is difficult because it seems so abstract. However, we should always bear in mind that thermodynamics is based on experimental observations. For instance, the First Law of thermodynamics expresses the em- pirical observation that energy is conserved, even though it can be converted in various forms. The internal energy of a system can be changed by performing work w on the system or by transferring an amount of heat q. It is meaning- less to speak about the total amount of heat in a system, or the total amount of work. This is not mysterious: it is just as meaningless to speak about the number of train-travelers and the number of pedestrians in a train-station: peo- ple enter the station as pedestrians, and exit as train-travelers or vice versa. However if we add the sum of the changes in the number of train-travelers and 1-2 pedestrians, then we obtain the change in the number of people in the station. And this quantity is well defined. Similarly, the sum of q and w is equal to the change in the internal energy U of the system dU = q + w (1.4) This is the First Law of Thermodynamics. The Second Law seems more ab- stract, but it is not. The Second Law is based on the experimental observation that it is impossible to make an engine that works by converting heat from a single heat bath (i.e. a large reservoir in equilibrium) into work. This observa- tion is equivalent to another - equally empirical - observation, namely that heat can never flow spontaneously (i.e. without performing work) form a cold reser- voir to a warmer reservoir. This statement is actually a bit more subtle than it seems because, before we have defined temperature, we can only distinguish hotter and colder by looking at the direction of heat flow. What the Second Law says is that it is never possible to make heat flow spontaneously in the “wrong” direction. How do we get from such a seemingly trivial statement to something as abstract as entropy? This is most easily achieved by introducing the concept of a reversible heat engine. A reversible engine is, as the word suggests, an engine that can be operated in reverse. During one cycle (a sequence of steps that is completed when the engine is returned into its original state) this engine takes in an amount of heat q1 from a hot reservoir converts part of it into work w and delivers a remaining amount of heat q2 to a cold reservoir. The reverse process is that, by performing an amount of work w, we can take an amount of heat q2 form the cold reservoir and deliver an amount of heat q1to the hot reservoir. Reversible engines are an idealization because in any real engine there will be additional heat losses. However, the ideal reversible engine can be approximated arbitrarily closely by a real engine if, at every stage, the real engine is sufficiently close to equilibrium. As the engine is returned to its original state at the end of one cycle, its internal energy U has not changed. Hence, the First Law tells us that dU = q (w + q ) = 0 (1.5) 1 − 2 or q1 = w + q2 (1.6) Now consider the “efficiency” of the engine η w/q1- i.e. the amount of work delivered per amount of heat taken in. At first,≡ one might think that η depends on the precise design of our reversible engine. However this is not true. η is the same for all reversible engines operating between the same two reservoirs. To demonstrate this, we show that if different engines could have different values for η then we would contradict the Second Law in its form “heat can never spontaneously flow from a cold to a hot reservoir”. Suppose therefore that we ′ have another reversible engine that takes in an amount of heat q1 form the hot reservoir, delivers the same amount of work w, and then delivers an amount of ′ ′ heat q2to the cold reservoir. Let us denote the efficiency of this engine by η . 1-3 Now we use the work generated by the engine with the highest efficiency (say η) to drive the second engine in reverse. The amount of heat delivered to the hot reservoir by the second engine is ′ ′ ′ q1 = w/η = q1(η/η ) (1.7) ′ ′ where we have used w = q1η. As, by assumption, η < η it follows that q1 > q1. Hence there is a net heat flow form the cold reservoir into the hot reservoir. But this contradicts the Second Law of thermodynamics. Therefore we must conclude that the efficiency of all reversible heat engines operating between the same reservoirs is identical. The efficiency only depends on the temperatures t1and t2 of the reservoirs (the temperatures t could be measured in any scale, e.g. in Fahrenheit 1 as long as heat flows in the direction of decreasing t). As η(t1,t2) depends only on the temperature in the reservoirs, then so does the ratio q2/q1 = 1 η. Let us call this ratio R(t2,t1). Now suppose that we have a reversible engine− that consists of two stages: one working between reservoir 1 and 2, and the other between 2 and 3. In addition, we have another reversible engine that works directly between 1 and 3. As both engines must be equally 1Daniel Gabriel Fahrenheit (1686 - 1736) was born in Danzig. He was only fifteen when his parents both died from eating poisonous mushrooms.
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