Anharmonic Oscillator Heat Capacity

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SUMMARY

The discussion focuses on deriving the heat capacity of a classical anharmonic oscillator described by the potential U(x) = cx² - gx³ - fx⁴. The approximate heat capacity is given by the formula C = kb[1 + (3f/2c² + 15g²/8c³)kbT]. Participants emphasize the importance of using the Boltzmann distribution and taking the derivative of the potential energy with respect to temperature to arrive at this result. A participant struggles to find the correct heat capacity, indicating a need for clarity in their calculations.

PREREQUISITES
  • Understanding of classical mechanics and potential energy functions
  • Familiarity with the Boltzmann distribution
  • Knowledge of heat capacity and its derivation
  • Basic calculus for differentiation
NEXT STEPS
  • Review the derivation of heat capacity from potential energy functions
  • Study the Boltzmann distribution in the context of statistical mechanics
  • Explore the implications of anharmonic potentials in thermodynamics
  • Investigate advanced topics in classical oscillators and their heat capacities
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Students and researchers in physics, particularly those studying thermodynamics and statistical mechanics, as well as anyone interested in the properties of anharmonic oscillators.

nyvane
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Homework Statement



Consider the anharmonic potential U(x)=cx2-gx3-fx4 and show that the approximate heat capacity of the classical unharmonic oscillator in one dimension is

C=kb[1+(3f/2c2+15g2/8c3)kbT]


Homework Equations



U(x)=cx2-gx3-fx4
and heat capacity is C=dU/dT

The Attempt at a Solution



I have used Boltzmann distrşbution of x as 3g/4c2*kbT and took derivative of U according to T but I could not find the given heat capacity.
 
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nyvane said:

Homework Statement



Consider the anharmonic potential U(x)=cx2-gx3-fx4 and show that the approximate heat capacity of the classical unharmonic oscillator in one dimension is

C=kb[1+(3f/2c2+15g2/8c3)kbT]

Homework Equations



U(x)=cx2-gx3-fx4
and heat capacity is C=dU/dT

The Attempt at a Solution



I have used Boltzmann distrşbution of x as 3g/4c2*kbT and took derivative of U according to T but I could not find the given heat capacity.

OK. But, I can't help you find your mistake if you don't show your calculations.
 
where did the 3g/4c2*kbT come from?
 

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