Second Law

1. Second Law We can imagine thermodynamic processes which conserve energy but which never occur in nature. For example, if we bring a hot object into contact with a cold object, we observe that the hot object cools down and the cold object heats up until an equilibrium is reached. The transfer of heat goes from the hot object to the cold object. We can imagine a system, however, in which the heat is instead transferred from the cold object to the hot object, and such a system does not violate the first law of thermodynamics. The cold object gets colder and the hot object gets hotter, but energy is conserved. Obviously we don't encounter such a system in nature and to explain this and similar observations, thermodynamicists (Clasius, Kelvin, and Carnot) proposed a second law of thermodynamics: Heat cannot spontaneously flow from a colder location to a hotter location In classical thermodynamics, the second law is a basic postulate applicable to any system involving heat energy transfer; in statistical thermodynamics, the second law is a consequence of the assumed randomness of molecular chaos. There are many versions of the second law, but they all have the same effect, which is to explain the phenomenon of irreversibility in nature. All complex natural processes are irreversible. The phenomenon of irreversibility results from the fact that if a thermodynamic system, which is any system of sufficient complexity, of interacting molecules is brought from one thermodynamic state to another, the configuration or arrangement of the atoms and molecules in the system will change in a way that is not easily predictable. A certain amount of "transformation energy" will be used as the molecules of the "working body" do work on each other when they change from one state to another. During this transformation, there will be a certain amount of heat energy loss or dissipation due to intermolecular friction and collisions; energy that will not be recoverable if the process is reversed. Nothing in life is certain except death, taxes and the second law of thermodynamics. All three are processes in which useful or accessible forms of some quantity, such as energy or money, are transformed into useless, inaccessible forms of the same quantity.

2. Irreversible processes are not inevitable, they are just overwhelmingly probable. Combinatorics and probability Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties. Examples: random walk, two-state systems, … Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur. An event (very loosely defined) – any possible outcome of some measurement. An event is a statistical (random) quantity if the probability of its occurrence, P, in the process of measurement is < 1. The “sum” of two events: in the process of measurement, we observe either one of the events. The “product” of two events: in the process of measurement, we observe both events. Independent events: one event does not change the probability for the occurrence of the other).

3. For one die, the probability of any face coming up is the same, 1/6. Therefore, it is equally probable that any number from one to six will come up. For two dice, what is the probability that the total will come up 2, 3, 4, etc. up to 12? List all possible outcomes (36) for a pair of dice. Total Combinations How Many 2 1+1 1 3 1+2, 2+1 2 4 1+3, 3+1, 2+2 3 5 1+4, 4+1, 2+3, 3+2 4 6 1+5, 5+1, 2+4, 4+2, 3+3 5

4. Microstates and Macrostates Each possible outcome is called a “microstate”. The combination of all microstates that give the same number of spots is called a “macrostate”. The macrostate that contains the most microstates is the most probable to occur. Throw two normal dice. What is the probability of two sixes coming up? When two dice are thrown, what is the probability of getting only one six? Probability of the six on the first die and not the second is: Probability of the six on the second die and not the first is the same, so:

5. Binomial Distribution Probability of n successes in N attempts: The total number of microstates is: Expectation value (or mean) of a macroscopic observable A (averaged over all accessible microstates s): Mean of the binomial distribution: Standard deviation: Standard deviation of the binomial distribution:

6. Toss 6 coins. Probability of n heads: For 1000 coins:

7. Two model systems with fixed positions of particles and discrete energy levels Both models are attractive because they can be described in terms of discrete microstates which can be easily counted (for a continuum of microstates, as in the example with a freely moving particle, we still need to learn how to do this). This simplifies calculation of. On the other hand, the results will be applicable to many other, more complicated models. Despite the simplicity of the models, they describe a number of experimental systems in a surprisingly precise manner. - two-state paramagnet .... (“limited” energy spectrum) - the Einstein model of a solid (“unlimited” energy spectrum)