## 经济代写|博弈论代写Game Theory代考|Temperature

If the expected value $\mu=E(v, \pi)$ is a measure for the degree of activity of $\mathfrak{S}$ relative to the probability distribution $\pi$ and the potential $v$, there may nevertheless be other probability distributions $\pi^{\prime}$ with the same expected value
$$E(v, \pi)=\mu=E\left(v, \pi^{\prime}\right) .$$
The idea is now to derive a canonical probability distribution $\beta$ with a given expected value $\mu$. To this end, we select $\beta$ as the distribution with the largest entropy:
$$H(\beta)=\max {H(\pi) \mid E(v, \pi)=\mu} .$$
As it turns out (Lemma $7.1$ below), every other probability distribution $\pi$ yielding the same expected value $\mu$ of $v$ will have a strictly smaller entropy $H(\pi)<H(\beta)$, which means that its specification would require more information. In this sense, $\beta$ is the unique “freest” (i.e., least biased) distribution with expectation $\mu$.

## 经济代写|博弈论代写Game Theory代考|BOLTZMANN distributions

Given the potential $v: \mathfrak{S} \rightarrow \mathbb{R}$ with values $v_\sigma=v(\sigma)$, any $t \in \mathbb{R}$ defines a related BoLTZMANN ${ }^1$ (probability) distribution $\beta(t)$ on $\mathfrak{S}$ with the components
$$\beta_\sigma(t)=\frac{e^{v_\sigma t}}{Z(t)} \quad \text { where } \quad Z(t)=\sum_{\sigma \in \mathfrak{S}} e^{v_\sigma t}>0 .$$
REMARK 7.1. The distributions $\beta(t)$ are also known as BoLTZMANN-GIBBS ${ }^2$ distributions. The function $Z(t)$ is the associated so-called partition function.

Computing derivatives, one finds 3 that the expected value of $v$ relative to $b(t)$ can be expressed as the logarithmic derivative of the partition function:
$$\mu(t)=\sum_{\sigma \in \mathfrak{S}} v_\sigma \beta_\sigma(t)=\frac{Z^{\prime}(t)}{Z(t)}=\frac{d \ln Z(t)}{d t} .$$
NoTA BENE. Under a BOLTZMANN distribution $\beta(t)$ with $t \in \mathbb{R}$, no state of $\mathfrak{S}$ will be impossible (i.e., occur with probability 0 ) and no state will occur with certainty (i.e., with probability 1 ) if $\mathfrak{S}$ has more than one state.

In the special case $t=0$, one has $Z(0)=|\mathfrak{S}|$. So $\beta(0)$ is the uniform distribution on $\mathfrak{S}$ with the average potential value as its expectation:
$$\mu(0)=\frac{1}{|\mathfrak{S}|} \sum_{\sigma \in \mathfrak{S}} v_\sigma .$$
Moreover, one observes the limiting behavior
$$\lim {t \rightarrow-\infty} \mu(t)=\min {\sigma \in \mathfrak{S}} v_\sigma \quad \text { and } \quad \lim {t \rightarrow+\infty} \mu(t)=\max {\sigma \in \mathfrak{S}} v_\sigma$$

## 经济代写|博弈论代写博弈论代考|温度

$$E(v, \pi)=\mu=E\left(v, \pi^{\prime}\right) .$$
。现在的想法是推导出一个规范概率分布$\beta$具有给定的期望值$\mu$。为此，我们选择$\beta$作为熵最大的分布:
$$H(\beta)=\max {H(\pi) \mid E(v, \pi)=\mu} .$$

## 经济代写|博弈论代写博弈论代考|BOLTZMANN分布

$$\beta_\sigma(t)=\frac{e^{v_\sigma t}}{Z(t)} \quad \text { where } \quad Z(t)=\sum_{\sigma \in \mathfrak{S}} e^{v_\sigma t}>0 .$$

$$\mu(t)=\sum_{\sigma \in \mathfrak{S}} v_\sigma \beta_\sigma(t)=\frac{Z^{\prime}(t)}{Z(t)}=\frac{d \ln Z(t)}{d t} .$$
NoTA BENE。在玻尔兹曼分布$\beta(t)$和$t \in \mathbb{R}$下，如果$\mathfrak{S}$有多个状态，则没有$\mathfrak{S}$的状态是不可能的(即，以概率0出现)，也没有状态是确定的(即，以概率1出现)

$$\mu(0)=\frac{1}{|\mathfrak{S}|} \sum_{\sigma \in \mathfrak{S}} v_\sigma .$$

$$\lim {t \rightarrow-\infty} \mu(t)=\min {\sigma \in \mathfrak{S}} v_\sigma \quad \text { and } \quad \lim {t \rightarrow+\infty} \mu(t)=\max {\sigma \in \mathfrak{S}} v_\sigma$$

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