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TA office hours: Switch to Thursdays 5 – 6 PM. Homework # 3 assigned today. Due next wed. The Canonical (Boltzmann) probability distribution “If we know the temperature of a system and the values of its external parameters, how can we estimate its physical properties, such as energy, pressure, magnetic moment, and distribution of molecular velocities? The question is answered {..} by deriving the canonical probability distribution…” R. Baierlein, Thermal Physics Meaning of probabilities in thermal physics Probabilities enter into thermal physics because the available data are insufficient to determine the individual properties of 1020 molecules. (Or, because of quantum mechanics, even the individual properties of a single molecule are expressed in terms of probabilities.) Even if all necessary data were available, no person or computer could cope with it. Meaning of probabilities in thermal physics Probabilities enter into thermal physics because the available data are insufficient to determine the individual properties of 1020 molecules. (Or, because of quantum mechanics, even the individual properties of a single molecule are expressed in terms of probabilities.) Even if all necessary data were available, no person or computer could cope with it. Probabilities always arise in a context. For example, the probability of 2 appearing if we role a die once is 1/6. Meaning of probabilities in thermal physics Probabilities enter into thermal physics because the available data are insufficient to determine the individual properties of 1020 molecules. (Or, because of quantum mechanics, even the individual properties of a single molecule are expressed in terms of probabilities.) Even if all necessary data were available, no person or computer could cope with it. Probabilities always arise in a context. For example, the probability of 2 appearing if we role a die once is 1/6. Two schools of thought on what probability means in thermal physics. Meaning of probability 1. Frequency meaning. A probability is a relative frequency in the long run, i.e. probability = (number of successes)/(number of tries) OR, probability = (number of objects with property X)/ (total number of objects) e.g. objects = 1020 gas of sodium atoms X = Na atom in first electronic excited state if this is true for 1016 atoms: Probability = 1016/1020 = 10-4 Meaning of probability 2. Degree of belief meaning. Probability is the rational degree of belief in the correctness of proposition A given the context B. For example: If A = “The first sodium atom we examine will be in the first excited electronic state.” and B = There exist 1020 Na gas atoms, 1016 of which are in the first excited state. Then, the rational degree of belief we can assign this statement A, given the context B, is 10-4. Why does this matter? (It does and it doesn’t.) “In thermal physics one often wants to estimate the behavior of a specific system, e.g. a lithium fluoride crystal grown last week in the lab which resides in the core of the labs only superconducting magnet. The degree of belief interpretation permits ‘one of a kind’ applications of probability theory. The frequency interpretation would require that one must imagine a large number of replicas of last week’s crystal and consider a relative frequency in that collection, or doing the same experiment on the same crystal again and again. Some physicists find such constructions artificial.” “Almost all calculations in thermal physics come out the same, Regardless of which interpretation of probability one adopts.” R. Baierlein Probabilities when the temperature is fixed How do we describe, in probabilistic terms, a system whose temperature is fixed? An example system from low temperature physics: Copper disk Cerium magnesium nitrate Cu cooled with liquid He which was pumped on for additional Evaporative cooling until all He evaporated. All isolated from enviroment In a cryostate (double walled vessel).