Fluctuations and Magnetism
As an introduction to the phenomena influenced by the effects of fluctuations, a brief explanation is given for the fluctuation-dissipation theorem and the Kramers-Kronig relation. As an example where fluctuations play the dominant role, we show how the t
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Fluctuations and Magnetism
2.1 Fluctuations Throughout this book, we are particularly concerned with the effects of fluctuations on various magnetic properties. As a brief introduction to the fluctuation phenomena, let us first take a system of a classical harmonic oscillator in equilibrium with its surroundings at temperature T . The Hamiltonian is given by H (q, p) =
1 1 2 p + V (q), V (q) = mω2 q 2 , 2m 2
(2.1)
where q and p represent a coordinate and its conjugate momentum. The mass of the particle and the vibration frequency are denoted by m and ω, respectively. When it is in thermal equilibrium, both of its variables q and p show random motions around the origin in the phase space. Deviations or fluctuations of variables are defined by δq ≡ q − q , δp ≡ p − p ,
(2.2)
where q and p are thermal averages of variables. Both of them are zero in this case. The variances are also defined for each variable by the average of fluctuation amplitude squared. δq 2 = q 2 − q 2 , δp 2 = p 2 − p 2
(2.3)
The above averages are easily evaluated as follows for the coordinate q: ∞
∞ δq = −∞ 2
dqd p q 2 e−H (q, p)/kB T
−H (q, p)/kB T −∞ dqd p e
=
kB T mω2
(2.4)
It corresponds to the law of equipartition of energy in classical statistical mechanics.
Y. Takahashi, Spin Fluctuation Theory of Itinerant Electron Magnetism, Springer Tracts in Modern Physics 253, DOI: 10.1007/978-3-642-36666-6_2, © Springer-Verlag Berlin Heidelberg 2013
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2 Fluctuations and Magnetism
In the presence of external force F in a positive direction, the potential energy V (q) of the system is then given by V (q) =
1 mω2 q 2 − Fq. 2
(2.5)
The stable position of the coordinate, shifted from the origin, is represented as follows. 1 q = χ F, χ = (2.6) mω2 The parameter χ defined as a coefficient of the F-linear term in the right hand side is generally called susceptibility. It characterizes the response of a system to the externally applied force. From the comparison of (2.4) and (2.6), it follows that the following relation is satisfied. (2.7) δq 2 = kB T χ The relation corresponds to the special case of the well-known fluctuation-dissipation theorem of statistical mechanics. It is the relation satisfied in general between the fluctuations and the response of the system to the external perturbation. In quantum mechanical treatment, it is better to introduce the two new variables b and b† by b=
mω 1 mω 1 † q +i p, b = q −i p. 2 2mω 2 2mω
(2.8)
Between them, the following commutation relation is satisfied. [b, b† ] = 1
(2.9)
The Hamiltonian is then represented by 1 , nˆ ≡ b† b. H = ω nˆ + 2
(2.10)
If we define the ground state by the condition bφ0 (q) = 0, excited eigenstates of n, ˆ φn (q), with integer eigenvalue n are successively generated by b† φn (q) =
√ n + 1φn+1 (q).
(2.11)
Thermal expectations of nˆ and q 2 are evaluated as follows. 1 2 n ˆ = ω/k T , q 2 = 1 + ω/k T B −1 B −1 2mω e e
(2.12)
2.1 Fluctuations
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It 2is easy to see that at hi
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