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Consider a system of two ising spin S1 and S2 taking value ±1 with interaction energy ∈ = -JS1S2 in thermal equilibrium at temperature T. The partition function of the system in the limit T → ∞ is ______.
Correct answer is '4'. Can you explain this answer?
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Consider a system of two ising spin S1and S2taking value ±1 wit...
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Consider a system of two ising spin S1and S2taking value ±1 wit...
+1 or -1. The energy of the system is given by:
E = -J(S1S2)
where J is a constant that represents the strength of the interaction between the spins.
The possible configurations of the system are:
S1 = +1, S2 = +1: energy E = -J
S1 = +1, S2 = -1: energy E = J
S1 = -1, S2 = +1: energy E = J
S1 = -1, S2 = -1: energy E = -J
The probability of each configuration depends on the temperature of the system and can be calculated using the Boltzmann distribution:
P(S1,S2) = exp(-E(S1,S2)/kT) / Z
where k is the Boltzmann constant, T is the temperature of the system, and Z is the partition function:
Z = ∑ exp(-E(S1,S2)/kT)
The partition function is the sum of the probabilities of all possible configurations and ensures that the probabilities are normalized.
The average value of each spin can be calculated as:
= ∑ S1P(S1,S2)
= ∑ S2P(S1,S2)
The correlation between the spins can be calculated as:
C = -
where is the average value of the product of the spins.
At high temperatures, when kT >> J, the Boltzmann distribution becomes nearly flat and all configurations are equally probable. In this case, the average value of each spin is zero and the correlation between the spins is also zero.
At low temperatures, when kT < j,="" the="" boltzmann="" distribution="" is="" dominated="" by="" the="" lowest="" energy="" configuration,="" where="" the="" spins="" are="" aligned.="" in="" this="" case,="" the="" average="" value="" of="" each="" spin="" is="" either="" +1="" or="" -1,="" depending="" on="" the="" sign="" of="" j,="" and="" the="" correlation="" between="" the="" spins="" is="" positive.="" j,="" the="" boltzmann="" distribution="" is="" dominated="" by="" the="" lowest="" energy="" configuration,="" where="" the="" spins="" are="" aligned.="" in="" this="" case,="" the="" average="" value="" of="" each="" spin="" is="" either="" +1="" or="" -1,="" depending="" on="" the="" sign="" of="" j,="" and="" the="" correlation="" between="" the="" spins="" is="" />
E = -J(S1S2)
where J is a constant that represents the strength of the interaction between the spins.
The possible configurations of the system are:
S1 = +1, S2 = +1: energy E = -J
S1 = +1, S2 = -1: energy E = J
S1 = -1, S2 = +1: energy E = J
S1 = -1, S2 = -1: energy E = -J
The probability of each configuration depends on the temperature of the system and can be calculated using the Boltzmann distribution:
P(S1,S2) = exp(-E(S1,S2)/kT) / Z
where k is the Boltzmann constant, T is the temperature of the system, and Z is the partition function:
Z = ∑ exp(-E(S1,S2)/kT)
The partition function is the sum of the probabilities of all possible configurations and ensures that the probabilities are normalized.
The average value of each spin can be calculated as:
The correlation between the spins can be calculated as:
C =
where
At high temperatures, when kT >> J, the Boltzmann distribution becomes nearly flat and all configurations are equally probable. In this case, the average value of each spin is zero and the correlation between the spins is also zero.
At low temperatures, when kT < j,="" the="" boltzmann="" distribution="" is="" dominated="" by="" the="" lowest="" energy="" configuration,="" where="" the="" spins="" are="" aligned.="" in="" this="" case,="" the="" average="" value="" of="" each="" spin="" is="" either="" +1="" or="" -1,="" depending="" on="" the="" sign="" of="" j,="" and="" the="" correlation="" between="" the="" spins="" is="" positive.="" j,="" the="" boltzmann="" distribution="" is="" dominated="" by="" the="" lowest="" energy="" configuration,="" where="" the="" spins="" are="" aligned.="" in="" this="" case,="" the="" average="" value="" of="" each="" spin="" is="" either="" +1="" or="" -1,="" depending="" on="" the="" sign="" of="" j,="" and="" the="" correlation="" between="" the="" spins="" is="" />
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Consider a system of two ising spin S1and S2taking value ±1 with interaction energy ∈ = -JS1S2 in thermal equilibrium at temperature T. The partition function of the system in the limit T → ∞ is ______.Correct answer is '4'. Can you explain this answer?
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