Theoretical prediction for absolute magnetization in 1D Ising model of a ferromagnet In the 2D Ising model of a ferromagnet, Onsager predicts the absolute magnetization as a function of (unitless) temperature as 
$$|M_{2D}(T)|=\left(1-\sinh\left(\frac{\ln\left(1+\sqrt{2}\right)}{T}\right)^{-4}\right)^\frac{1}{8}$$
Does a similar expression for the absolute magnetization exist for the 1D model? I am able to find a theoretical prediction for the non-absolute magnetization in the presence of an external magnetic field $B$ (in unitless variables),
$$M_{1D}(T)=\frac{\exp\left(\frac{1}{T}\right)\sinh(B)}{\sqrt{\left(\exp\left(\frac{2}{T}\right)\sinh^2(B)+\exp{\left(-\frac{2}{T}\right)}\right)}},$$
but this vanishes when $B=0$, due to the symmetry of the system. I have so far been unable to find an analytical expression for $|M_{1D}(T)|$. Does such a thing exist?
 A: Let
$$
m_N = \frac1N \sum_{i=1}^{N} \sigma_i
$$
be the (empirical) magnetization density and $\mu_{N;\beta}$ be the Gibbs measure (with periodic boundary condition) at inverse temperature $\beta<\infty$, for a one-dimensional system of $N$ spins $\sigma_1,\ldots,\sigma_N$.
One can then prove the following: For any $\beta<\infty$ and any $\epsilon>0$,
$$
\lim_{N\to\infty} \mu_{N;\beta}(|m_N| > \epsilon) = 0.
$$
(In fact, this probability decays exponentially fast in $N$.) This is standard material (for a proof, see for example Theorem 3.10 in this book). Of course, it follows that the expected value of $|m_N|$ satisfies
$$
\langle |m_N| \rangle_{N;\beta} \leq \epsilon + \mu_{N;\beta}(|m_N| > \epsilon).
$$
In view of the above, this immediately implies that
$$
\lim_{N\to\infty} \langle |m_N| \rangle_{N;\beta} = 0,
$$
for any $\beta<\infty$.
A: I found a reasonable approximation here, in equation 
2.36. Getting the absolute magnetization amounts to solving the implicit equation
$$M=\tanh\left(\frac{|M|}{T}\right)$$
