( the best time)
Physics time
Why do we care about Statistical Mechanics?
It relates the
microscopic
with the
MACROSCOPIC
( the best time)
Physics time
Why do we care about Statistical Mechanics?
It relates the
microscopic
with the
MACROSCOPIC
( the best time)
Why do we care about Statistical Mechanics?
Physics time
An ising model recap:
part 2
Ludmila Augusta Soares Botelho
Physics time
$$(x,y,z)$$
$$m$$
$$m$$
$$\vec{p}$$
$$(x,y,z, p_x,p_y,p_z)$$
$$(x,y,z)$$
$$\vec{v}$$
Statistical mechanics
$$m$$
$$(x,y,z, p_x,p_y,p_z)$$
- Microstate
Statistical mechanics
- Microstate
- For particles:
Statistical mechanics
- Microstate
- For particles:
Statistical mechanics
- Microstate
- For particles:
Statistical mechanics
- Microstate
- For particles:
thermodynamic Quantities
- Macrostate
- Equilibrium observable state
microstate
thermodynamic Quantities
- Macrostate
Condensed matter physics
The bridge
$$\Omega(E) =\text{Number of Microstates} $$
$$S = -k_b\sum_i p_i \ln p_i$$
- Entropy
$$S = k_b\ln W$$
$$W = \frac{N!}{\prod_i N_i }$$
$$S = k_b\ln \Omega$$
- Gibbs Entropy
- Postulate: All microstates are equally probable
$$S = -k_b\sum_i^\Omega p_\mu \ln p_\mu$$
$$p(\mu)= \frac{1}{\Omega(E)}$$
$$ = -k_b\sum_i^\Omega \frac{1}{\Omega (E)} \ln \frac{1}{\Omega (E)}$$
$$= - k_b\ln \frac{1}{\Omega (E)}$$
$$S = k_b\ln {\Omega (E)}$$
The bridge
- Entropy
$$S = k_b\ln \Omega$$
- State Function/Fundamental Equation
$$S = S(U,V,N)$$
$$ \frac{1}{T}= \frac{\partial S}{\partial U}$$
$$ \frac{p}{T}= \frac{\partial S}{\partial U}$$
$$ \frac{-\mu}{T}= \frac{\partial S}{\partial U}$$
temperature
pressure
chemical potential
The bridge
- Helmholtz Free Energy
$$F= U - TS$$
$$U =-\vec{\mu} \vec{H}$$
$$m =-\left. \left( \frac{\partial F}{\partial H} \right)\right |_{T,N}$$
$$z$$
$$\vec{H}$$
- Legendre Transformation for internal energy \(U = Q +W \)
$$\chi (T,H) =-\left. \left( \frac{\partial F}{\partial H} \right)\right |_{T,N}$$
$$U =-\mu_0 {H}$$
$$ \rightarrow \frac{CH}{T}$$
$$M =N\langle{m} \rangle$$
$$\mathrm{d}F= M\mathrm{d}H - S\mathrm{d}T$$
Probability DISTRIBUTION
with the
MACROSCOPE
Canonical Ensemble
S
R
~
~
~
(T, V, N)
= E
total
$$E_ = E_S + E_R$$
$$S(E_R) =k_b \ln \Omega (E_R)$$
$$ \Omega(E_R) = \exp\left[\frac{S(E_R)}{k_b} \right]$$
$$ P(E_S)\propto \Omega(E - E_S)$$
$$E_R = E-E_S$$
$$ P_j= \alpha \Omega(E - E_j)$$
$$ \ln P_j= \ln \alpha + \ln \Omega(E) + \left[ \frac{\partial \ln \Omega(E_R)}{\partial E}\right]_{E_R=E} (-E_j)$$
$$+ \frac{1}{2}\left[ \frac{\partial^2 \ln \Omega(E)}{\partial E^2}\right]_{E_R=E}(-E_j)^2 +\cdots$$
$$\frac{1}{k_B}\frac{\partial S}{\partial U}=\frac{1}{k_BT}$$
$$\frac{1}{k_B}\frac{\partial^2 S}{\partial U^2}=\frac{1}{k_B}\frac{\partial}{\partial U}\left(\frac{1}{T}\right)$$
$$ \rightarrow 0$$
$$ \ln P_j= c -\frac{1}{k_b T}E_j$$
$$ P_j= \frac{\exp(-\beta E_j)}{\sum_k \exp(-\beta E_j)}$$
partition function
$$\sum_j E_j P_j = \langle E \rangle = U$$
$$\sum_j P_j = 1$$
$$ f(\{P_j\}, \lambda_1, \lambda_2)= -k_b\sum_j P_j \ln P_j - \lambda_1 \left( \sum_j P_j -1 \right)$$
$$ - \lambda_2 \left( \sum_j E_jP_j -U \right)$$
$$\frac{\partial f}{\partial P_j}= -k_b P_j \ln P_j -k_B - \lambda_1 -\lambda_2 E$$
$$ P_j= \frac{\exp(-\beta E_j)}{\sum_k \exp(-\beta E_j)}$$
Partition Function
- Thermodynamic Limit
$$ F = -\frac{1}{\beta} \ln Z$$
$$ Z \rightarrow \exp (-\beta F)$$
$$ Z = \sum_j \exp(-\beta E_j)$$
$$ = \sum_E \exp[-\ln\Omega(E)\beta E]$$
$$ Z \sim \exp[- \min\{E -TS(E)\}$$
$$S(E) =k_b \ln \Omega (E)$$
quantum mechanics
$$\mathcal{H} = -J \sum_{i} s_i s_{i+1} - h\sum_i^N s_i$$
$$\mathcal{H} = -J \sum_{i} \sigma_i^z \sigma_{i+1}^z $$
Pauli Matrices
Reduces to classical version
$$\mathcal{H} = -J \sum_{i} \sigma_i^z \sigma_{i+1}^z - h\sum_i^N \sigma_i^z$$
$$E = -J \sum_{i} \sigma_i \sigma_{i+1} $$
transverse field
energy levels \( \sigma_i =\pm 1 \)
diagonal!
- It has phase transition!
-Ferromagnetic vs Paramagnet
$$ \sigma_i^\alpha = I^{\otimes i-1} \otimes \sigma^\alpha I^{\otimes N-i} $$
$$\mathcal{H} = -J \sum_{i} \sigma_i^z \sigma_{i+1}^z - h\sum_i^N \sigma_i^x$$
quantum mechanics
$$\mathcal{H} = -J \sum_{i} \sigma_i^z \sigma_{i+1}^z - h\sum_i^N \sigma_i^x$$
$$\mathcal{H} = - \frac{1}{2} \sum_{i}^N \left[ J_x \sigma_i^x \sigma_{i+1}^x +J_y \sigma_{i}^y \sigma_{i+1}^y+ J_z \sigma_i^z \sigma_{i+1}^z - h\sigma_i^z\right]$$
$$\mathcal{H} = - \frac{1}{2} \sum_{i}^N \left[ J_x \sigma_i^x \sigma_{i+1}^x +J_y \sigma_{i}^y \sigma_{i+1}^y+ - h\sigma_i^z\right]$$
$$J_x = \left( \frac{1+ \gamma}{2} \right)$$
$$J_y = \left( \frac{1- \gamma}{2} \right)$$
- Tensor Networks
- Quantum Many-Body Problems
- Heisenberg Model
- XY Model
- 1D Quantum Ising Model
$$\mathcal{H} = -J \sum_{i} \sigma_i^z \sigma_{i+1}^z - h\sum_i^N \sigma_i^x$$
$$J > 0$$
$$J < 0$$
$$J = ?$$
$$\mathcal{H} = -J \sum_{i} \sigma_i^z \sigma_{i+1}^z$$
Ground state
Spin Glass Model
$$\mathcal{H} = - \sum_{i,j} J_{i,j}\sigma_i^z \sigma_{j}^z$$
magnet