Statistical Mechanics Lecture 7

The speaker begins by expressing their intention to discuss the second law of thermodynamics, emphasizing its importance and complexity. They mention that understanding this law is crucial and challenging, suggesting that very few people have a deep grasp of it. Before diving into the law, the speaker wants to explore physical examples related to statistical mechanics, specifically focusing on the speed of sound in gases. They propose a thought experiment involving a dilute gas and ponder the speed at which sound travels through it, relating it to the average velocity of gas molecules.

The discussion shifts to the speed of sound in a gas, particularly a dilute one. Two formulas are alluded to, but the focus is on the behavior of sound in a very dilute gas. The speaker theorizes that the speed of sound might be close to the average velocity of the molecules. To support this claim, they derive a formula for the average velocity of molecules in a gas at thermal equilibrium at a given temperature T. The relationship between temperature, kinetic energy, and the mass of the molecules is explored, leading to an estimation of the speed of sound that is consistent with known values.

The speaker poses a puzzle regarding the constancy of temperature in relation to the speed of sound. They assume that for small amplitude sound waves, the temperature remains fairly constant, which is a reasonable assumption for an ideal gas. They acknowledge some uncertainty about this assumption but suggest it's likely accurate. The conversation then turns to the relationship between pressure variation, density, and the speed of sound. It's noted that while pressure and density variations are small, their derivative with respect to each other is not necessarily small, hinting at a complex relationship in non-ideal conditions.

The topic of a harmonic oscillator is introduced, moving from the study of gases to oscillatory systems. The speaker describes a harmonic oscillator as a system that oscillates when disturbed, such as a spring-mass system or an electromagnetic wave in a cavity. They aim to calculate the average energy of the oscillator and how it depends on mass and the spring constant. The energy of the oscillator is expressed in terms of its coordinate and the spring constant. The speaker then outlines the process of applying statistical mechanics to find the average energy of the oscillator by using the Boltzmann distribution.

The speaker delves into the calculation of the partition function for a quantum harmonic oscillator. They explain that the partition function is a crucial concept in statistical mechanics and express it as an integral over momentum and position space. The integration process is detailed, highlighting the separation of variables and the use of the Gaussian integral result. The partition function is simplified to an expression involving fundamental constants, the mass of the oscillator, and the spring constant. The energy of the oscillator is then derived from the partition function, revealing that the average energy depends on temperature but not on the mass or spring constant.

The speaker contrasts the classical and quantum mechanical treatment of a harmonic oscillator. They highlight the paradox that in the classical view, the energy of an oscillator is independent of its mass and spring constant, which seems counterintuitive. The speaker suggests that this paradox was a point of confusion historically and led to the realization that classical physics was incomplete. They introduce quantum mechanics as the missing ingredient, explaining that the energy levels of an oscillator are quantized in quantum mechanics. The quantum mechanical partition function is derived, and the energy calculation reflects the discrete energy levels of the oscillator.

The discussion turns to the specific heat of solids, which was a problem that classical physics could not accurately explain. The speaker credits Einstein for proposing that quantum mechanics could resolve the discrepancy by showing that at low temperatures, the energy levels of oscillators in a crystal are suppressed, leading to lower specific heat than predicted by classical physics. The speaker emphasizes that quantum mechanics provides a correction to the classical view and that the transition from quantum to classical behavior in oscillators occurs when the temperature is high enough to provide more than one quantum's worth of energy.

The speaker explores the concept of chaotic systems and their relation to the second law of thermodynamics. They describe how small differences in initial conditions can lead to vastly different outcomes in chaotic systems, making long-term predictions impossible without infinite precision. This sensitivity to initial conditions and the resulting unpredictability are central to the idea that entropy, or the degree of disorder in a system, tends to increase over time. The speaker also touches on the historical significance of understanding entropy and the second law, mentioning the contributions of Boltzmann and the concept of coarse-graining in phase space.

The speaker addresses the paradox between the reversibility of Newton's laws of motion and the apparent irreversibility of macroscopic phenomena, as described by the second law of thermodynamics. They discuss how the concept of coarse-graining in phase space helps to resolve this paradox by introducing a measure of uncertainty that aligns with real-world observations. The speaker suggests that while a system can return to its original configuration, the likelihood of this happening becomes increasingly small as the system size grows, reflecting the practical irreversibility observed in nature.

The speaker concludes with a discussion on the nature of chaos in dynamical systems. They define chaos as a situation where nearby trajectories in phase space diverge exponentially over time, leading to a loss of predictability. They contrast this with non-chaotic systems, such as a simple pendulum or an orbit around the Sun, where nearby trajectories remain close. The speaker acknowledges the complexity of determining whether a given system is chaotic and notes that while most systems are unpredictable over time due to chaos, knowing the initial conditions with infinite precision would allow for perfect predictability.