Glasses are materials that are disordered – and not crystalline – at low temperatures. Here is a simple model. Consider a system with energy E, the number of accessible microstates is given by a Gaussian function: Ω(E) = Ω0 e −(E−E¯) 2/(2∆2 ) , where Ω0 and E¯ and ∆ are positive constants. E¯ is the average energy. In this problem, we consider only the states whose energy E is below E¯. 1. Show that the entropy is an inverted parabola: S(E) = S0 − α (E − E¯) Find S0 and α. Write your answers in terms Ω0, ∆, and other universal constants. 2. An entropy catastrophy happens when S = 0, which occurs at energy E0. (i) Find E0. (ii) What is the number of accessible states for energy below E0? 3. The glass transition temperature Tg is the temperature of entropy catastrophy. Compute Tg.
Glasses are materials that are disordered – and not crystalline – at low temperatures. Here is a simple model. Consider a system with energy E, the number of accessible microstates is given by a Gaussian function: Ω(E) = Ω0 e −(E−E¯) 2/(2∆2 ) , where Ω0 and E¯ and ∆ are positive constants. E¯ is the average energy. In this problem, we consider only the states whose energy E is below E¯.
1. Show that the entropy is an inverted parabola: S(E) = S0 − α (E − E¯) Find S0 and α. Write your answers in terms Ω0, ∆, and other universal constants.
2. An entropy catastrophy happens when S = 0, which occurs at energy E0. (i) Find E0. (ii) What is the number of accessible states for energy below E0?
3. The glass transition temperature Tg is the temperature of entropy catastrophy. Compute Tg.
4. Find the energy E as a function of T.
5. Without any calculation, explain why E → E¯ as T → +∞. Hint: consider the Boltzmann distribution
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