Stability, Emergence and Part-Whole Reduction

Abstract

Often we can describe the macroscopic behaviour of systems without knowing much about the nature of the constituents of the systems let alone the states the constituents are in. Thus, we can describe the behaviour of real or ideal gases without knowing the exact velocities or places of the constituents. It suffices to know certain macroscopic quantities in order to determine other macroscopic quantities. Furthermore, the macroscopic regularities are often quite simple. Macroscopic quantities are often determined by only a few other macroscopic quantities.

Appendices

Renormalization and Cumulant Generating Functions

The renormalization group transformation for the case of sums of independent random variables is best investigated in terms of their cumulant generating functions.

Given a random variable $X$, its characteristic function is defined as the Fourier transform of its probability density $p_{X}$,¹¹

$$ \varphi_{X} (k) = \left\langle {{\text{e}}^{ikX} } \right\rangle = \int {\text{d}}x\;p_{X} (x)\;{\text{e}}^{ikx} . $$

(10.19)

Characteristic functions are important tools in probability. Among other things, they can be used to express moments of a random variable in compact form via differentiation,

$$ \left( { - i\frac{\text{d}}{{{\text{d}}k}}} \right)^{n} \varphi_{X} (k)|_{k = 0} = \left\langle {X^{n} } \right\rangle = \int {\text{d}}x\;p_{X} (x)\;x^{n} . $$

(10.20)

For this reason, characteristic functions are also referred to as moment generating functions. A second important property needed here is that the characteristic function of a sum $X + Y$ of two independent random variables $X$ and $Y$ is given by the product of their characteristic functions. For, denoting by $p_{X}$ and $p_{Y}$ the probability densities corresponding to the two variables, one finds

$$ \varphi_{X + Y} (k) = \int {\text{d}}x{\text{d}}y\;p_{X} (x)p_{Y} (y)\;{\text{e}}^{ik(x + y)} = \varphi_{X} (k)\varphi_{Y} (k). $$

(10.21)

Rather than characterizing random variables in terms of their moments, it is common to use an equivalent description in terms of so-called cumulants instead. Cumulants are related to moments of centered distributions and can be seen to provide measures of “dispersion” of a random variable. They are defined as expansion coefficients of a cumulant-generating function (CGF) which is itself defined as the logarithm of the moment generating function

$$ f_{X} (k) = \log \varphi_{X} (k); $$

(10.22)

hence $n$-th order cumulants $\kappa_{n} (X)$ are given by

$$ \kappa_{n} (X) \equiv \left( { - i\frac{\text{d}}{{{\text{d}}k}}} \right)^{n} f_{X} (k)|_{k = 0} ,\quad n \ge 1. $$

(10.23)

The two lowest order cumulants are $\kappa_{1} (X) = \mu$, and $\kappa_{2} (X) = {\text{Var}}(X)$, i.e., the mean and the second centered moment. The multiplication property of characteristic functions of sums of independent random variables translates into a corresponding addition property of the CGF of sums of independent variables,

$$ f_{X + Y} (k) = f_{X} (k) + f_{Y} (k), $$

(10.24)

entailing that cumulants of sums of independent random variables are additive.

We note in passing that characteristic functions are the probabilistic analogues of partition functions in Statistical Mechanics, and hence that cumulant generating functions are probabilistic analogues of free energies.

We now proceed to use the additivity relations of CGFs to investigate the properties of sums of random variables (10.5),

$$ S_{N} (\varvec{s}) = \frac{1}{{N^{1/\alpha } }}\sum\limits_{k = 1}^{N} s_{k} $$

under renormalization. We denote CGF corresponding to $S_{N}$ by $F_{N}$. Let $f_{1}$ denote the CGF of the original variables, $f_{2}$ that of the renormalised variables $s_{k}^{\prime }$ (constructed from sums of two of the original variables), and more generally, let $f_{{2^{\ell } }}$ denote the CGF of the $\ell$-fold renormalised variables, constructed from sums involving $2^{\ell }$ original variables. We then get

$$ \begin{aligned} F_{N} (k): & = \log \left\langle {\text{e}^{{ikS_{N} }} } \right\rangle = Nf_{1} \left( {\frac{k}{{N^{1/\alpha } }}} \right) = \frac{N}{2}f_{2} \left( {\frac{k}{{(N/2)^{1/\alpha } }}} \right) \\ & = \frac{N}{4}f_{4} \left( {\frac{k}{{(N/4)^{1/\alpha } }}} \right) = \ldots = \frac{N}{{2^{\ell } }}f_{{2^{\ell } }} \left( {\frac{k}{{(N/2^{\ell } )^{1/\alpha } }}} \right). \\ \end{aligned} $$

(10.25)

Assuming that multiply renormalised variables will acquire asymptotically stable statistical properties, i.e. statistical properties that remain invariant under further renormalization, the $f_{{2^{\ell } }}$ would have to converge to a limiting function $f^{*}$,

$$ f_{{2^{\ell } }} \to f^{*} ,\quad \ell \to \infty . $$

(10.26)

This limiting function $f^{*}$ would have to satisfy a functional self-consistency relation of the form

$$ f^{*} (2^{1/\alpha } k) = 2f^{*} (k) $$

(10.27)

which follows from (10.25). This condition states that the invariant CGF $f^{*} (k)$ must be a homogeneous function of degree $\alpha$.

The solutions of this self-consistency relation for $\alpha = 1$ and $\alpha = 2$ are thus seen to be given by

$$ f^{*} (k) \equiv \mathop {\lim }\limits_{N \to \infty } F_{N} (k) = c_{\alpha } k^{\alpha } , $$

(10.28)

One identifies the CGF of a non-fluctuating (i.e. constant) random variable with $c_{\alpha } = i\langle X\rangle = i\mu$ for $\alpha = 1$, and that of a Gaussian normal random variable with zero-mean and variance $\sigma^{2}$ with $c_{\alpha } = - \frac{1}{2}\sigma^{2}$ for $\alpha = 2$, and thereby verifies the statements of the two limit theorems.

One can also show that the convergence (10.26) is realised for a very broad spectrum of distributions for the microscopic variables, both for $\alpha = 1$ (the law of large numbers), and for $\alpha = 2$ (the central limit theorem). For $\alpha = 1$, there is a “marginal direction” in the infinite-dimensional space of possible perturbations of the invariant CGF (corresponding to a change of the expectation value of the random quantities being summed), which doesn’t change its distance to the invariant function $f^{*} (k)$ under renormalization. All other perturbations are irrelevant in the sense that their distance from the invariant CGF will diminish under repeated renormalization. For $\alpha = 2$ there is one “relevant direction” in the space of possible perturbations, in which perturbations of the invariant CGF will be amplified under repeated renormalization (it corresponds to introducing a non-zero mean of the random variables being added), and a marginal direction that corresponds to changing the variance of the original variables. All other perturbations are irrelevant and will be diminished under renormalization. The interested reader will find a formal verification of these statements in the following Appendix 10.9. Interestingly that stability analysis will also allow to quantify the rate of convergence to the limiting distribution as a function of system size $N$ (for sums of independent random variables which—apart from having finite cumulants—are otherwise arbitrary).

Linear Stability Analysis

Statements about the stability of invariant CGF under various perturbations are proved by looking at the linearisation of the renormalization group transformation in the vicinity of the invariant CGF. We shall see that this description considerably simplifies the full analysis compared to the one in terms of probability densities used in (Sinai 1992).

Let $R_{\alpha }$ denote the renormalization transformation of a CGF for the scaling exponent $\alpha$. From (10.25), we see that its action on a CGF $f$ is defined as

$$ R_{\alpha } [f](2^{1/\alpha } k) = 2f(k). $$

(10.29)

Assuming $f = f^{*} + h$, where $h$ is a small perturbation of the invariant CGF, we have

$$ R_{\alpha } [f^{*} + h](2^{1/\alpha } k) = 2(f^{*} (k) + h(k)). $$

(10.30)

Using an expansion of the transformation $R_{\alpha }$ in the vicinity of $f^{*}$, and denoting by $D_{\alpha } = D_{\alpha } [f^{*} ]$ the operator of the linearised transformation in the vicinity of $f^{*}$ on the l.h.s, one has $R_{\alpha } [f^{*} + h] \simeq R_{\alpha } [f^{*} ] + D_{\alpha } h$ to linear order in $h$, thus

$$ R_{\alpha } [f^{*} ](2^{1/\alpha } k) + D_{\alpha } h(2^{1/\alpha } k) \simeq 2f^{*} (k) + 2h(k). $$

(10.31)

By the invariance of $f^{*}$ under $R_{\alpha }$, we get

$$ D_{\alpha } h(2^{1/\alpha } k) = 2h(k) $$

(10.32)

to linear order. The stability of the invariant CGF is then determined by the spectrum of $D_{\alpha }$, found by solving the eigenvalue problem

$$ D_{\alpha } h(2^{1/\alpha } k) = 2h(k) = \lambda h(2^{1/\alpha } k). $$

(10.33)

Clearly this equation is solved by homogeneous functions:

$$ h(k) = h_{n} (k) = \kappa_{n} \frac{{(ik)^{n} }}{n!}, $$

(10.34)

for which

$$ 2h_{n} (k) = \lambda_{n} h_{n} (2^{1/\alpha } k) $$

entails

$$ \lambda_{n} = 2^{1 - n/\alpha } . $$

(10.35)

In order for $f^{*} + h_{n}$ to describe a system with finite cumulants, we must have $n \ge 1$.

For the case $\alpha = 1$ then we have $\lambda_{1} = 1$ (the corresponding perturbation being marginal), and $\lambda_{n} < 1$ for $n > 1$ (the corresponding perturbations thus being irrelevant). The marginal perturbation amounts to changing the mean of the random variable to $\mu + \kappa_{1}$, as mentioned earlier.

In the case where $\alpha = 2$ we have that $\lambda_{1} = 2^{\frac{1}{2}}$ (the corresponding perturbation being relevant), $\lambda_{2} = 1$ (the corresponding perturbation being marginal), and $\lambda_{n} < 1$ for all $n > 2$ (the corresponding perturbations thus being irrelevant). The relevant perturbation amounts to introducing a nonzero mean $\mu = \kappa_{1}$ of the original random variables, while the marginal perturbation changes the variance to $\sigma^{2} + \kappa_{2}$, as mentioned earlier. All other perturbations change higher order cumulants of the random variables considered and are irrelevant.

Knowledge about the eigenfunctions of the linearized RG transformation and their eigenvalues allows to obtain a complete overview over the finite $N$ corrections to the limit theorems we have looked at in Appendix 10.8. Suppose we have

$$ f_{1} (k) = f^{*} (k) + \sum\limits_{n > 1} h_{n} (k) = f^{*} (k) + \sum\limits_{n > 1} \kappa_{n} \frac{{(ik)^{n} }}{n!} $$

in (10.25). Then after $\ell$ coarse-graining steps we have

$$ \begin{aligned} F_{N} (k) & = Nf_{1} \left( {\frac{k}{{N^{1/\alpha } }}} \right) = f^{*} (K) + N\sum\limits_{n > 1} h_{n} \left( {\frac{k}{{N^{1/\alpha } }}} \right) \\ & = f^{*} (K) + \frac{N}{{2^{\ell } }}\sum\limits_{n > 1} \lambda_{n}^{\ell } h_{n} \left( {\frac{k}{{(N/2^{\ell } )^{1/\alpha } }}} \right) \\ \end{aligned} $$

(10.36)

where we have exploited the invariance and homogeneity of $f^{*} (k)$, and the fact that each coarse graining step rescales the eigenfunction $h_{n}$ by an eigenvalue $\lambda_{n}$. We have recorded this relation in a slightly more complicated version than necessary to formally link it up with the analogous steps used in the derivation of finite-size scaling relations in the case of interacting systems. The reader is invited to check correctness of (10.36) herself using nothing but the homogeneity of the $h_{n}$.

In a system with finite $N$, the number $\ell$ of coarse graining steps that can be performed is necessarily finite, and in fact restricted to $2^{{\ell_{{{\text{m}}ax}} }} = N$. Using this maximum value in (10.36), we get

$$ F_{N} (k) = f^{*} (K) + \sum\limits_{n > 1} \lambda_{n}^{{\ell_{\hbox{max} } }} h_{n} (k) = f^{*} (K) + \sum\limits_{n > 1} N^{1 - n/\alpha } h_{n} (k) $$

(10.37)

which is the result that would have been obtained by just using homogeneity in $F_{N} (k) = Nf_{1} \left( {\frac{k}{{N^{1/\alpha } }}} \right)$. This result entails that higher order cumulants of $S_{N}$ scale with inverse powers of $N$ according to

$$ \kappa_{n} (S_{N} ) = N^{1 - n/\alpha } \kappa_{n} (S_{1} ), $$

(10.38)

so that, e.g., all cumulants higher than the second order cumulant will vanish in the infinite system limit in the case $\alpha = 2$ of the central limit theorem.