In the canonical ensemble, the energy fluctuations ΔE are given by the equation (ΔE)² = kBT² (∂E(T,x)/∂T), as stated in problem 19.1. Here's a step-by-step explanation:
1. Start with the definition of energy fluctuations: ΔE = E - ⟨E⟩, where E is the energy of a particular microstate and ⟨E⟩ is the average energy averaged over all microstates.
2. To find the energy fluctuations, we need to calculate the variance of energy, which is given by Var(E) = ⟨(E - ⟨E⟩)²⟩, where ⟨⟩ denotes the average over all microstates.
3. Using statistical mechanics, we can relate the variance of energy to the partial derivative of average energy with respect to temperature: Var(E) = kBT² (∂E/∂T).
4. The energy fluctuations ΔE are the square root of the variance, so we have ΔE = √(Var(E)) = √(kBT² (∂E/∂T)).
Thus, we have shown that the energy fluctuations ΔE in the canonical ensemble are given by (ΔE)² = kBT² (∂E(T,x)/∂T), as required in problem 19.1.
Moving on to problem 19.2, we need to show that the heat capacity CV = (∂E(T,V,N)/∂T) is always positive. Here's an explanation:
1. The heat capacity CV measures how the energy of a system changes with temperature. It is defined as CV = (∂E/∂T) at constant volume V and particle number N.
2. A positive heat capacity indicates that an increase in temperature leads to an increase in the energy of the system, while a negative heat capacity implies the opposite.
3. In thermodynamics, it can be proven that the heat capacity at constant volume is always positive for any system.
Therefore, we can conclude that the heat capacity CV = (∂E(T,V,N)/∂T) has to be positive, as stated in problem 19.2.
Lastly, problem 19.3 asks us to justify the statement ΔE/E ∼ 1/√N. Here's an explanation:
1. Recall that ΔE represents the energy fluctuations and E is the average energy.
2. In the canonical ensemble, the energy fluctuations ΔE scale with the square root of the number of particles N.
3. As the number of particles increases, the fluctuations in energy become relatively smaller compared to the average energy. Thus, ΔE/E ∼ 1/√N.
To summarize, in the canonical ensemble, the energy fluctuations are given by (ΔE)² = kBT² (∂E(T,x)/∂T) (problem 19.1). The heat capacity CV = (∂E(T,V,N)/∂T) is always positive (problem 19.2). The statement ΔE/E ∼ 1/√N is justified by the scaling of energy fluctuations with the square root of the number of particles (problem 19.3).