Relative error definition of entropy

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SUMMARY

The discussion focuses on the relative error in the definition of entropy when comparing two systems in thermal contact, specifically when one system is significantly larger than the other. The key equation discussed is the approximation of entropy as ln(g1g2)max, which is valid when both systems contain a large number of particles. In the case presented, with N1 = 10E22 spins and N2 = 10 spins, the relative error in entropy calculation is minimal, affirming the use of this approximation under the condition that N1 >> N2.

PREREQUISITES
  • Understanding of statistical mechanics and entropy concepts
  • Familiarity with the properties of large numbers in thermodynamics
  • Knowledge of the function g(N,U) and its significance in entropy calculations
  • Basic proficiency in logarithmic functions and their applications in physics
NEXT STEPS
  • Study the properties of entropy in statistical mechanics
  • Learn about the function g(N,U) and its derivation
  • Explore the implications of the additivity property of entropy
  • Investigate the concept of relative error in thermodynamic equations
USEFUL FOR

This discussion is beneficial for students and professionals in physics, particularly those studying statistical mechanics, thermodynamics, or anyone involved in advanced entropy calculations and their applications in physical systems.

kraigandrews
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Homework Statement


Problem-5: In the class we saw that property of additivity of the entropy for systems in thermalcontact relies on the approximation that we compute entropy of the most probable state ln(g1g2)maxinstead of full ln g(N,U) = ln(P,U1)
g(N1,U1)g(N2,U −U1)). The error made turns out to be really
small provided that each systems contains large number of particles. For the example of spins with
N1 = N2 = N/2 >> 1 we found
ln g(N, s) = ln(g1g2)max +1/2ln(N/8).
(4)
Consider now a special case when one system is large having N1 = 10E22 spins but the other is relatively
small with only N2 = 10 spins. Take s = 0 and compute the relative error in the definition
of entropy. Is it still a good idea to use such definition of entropy when N1 >> N2? [1 pt]


Homework Equations





The Attempt at a Solution


Ok so I honestly haven't a clue where to start I am just looking for type of guidance. My first idea would be to find g1 and g2 then from there I am totally lost.
 
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I know that g(N,U) = ln(P,U1) g(N1,U1)g(N2,U −U1)). But I don't know how to use it or really what it means.
 

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