# An idea to model a one-dimensional thermometer?

Let's say I have a $$1$$-dimensional material with thermal expansion $$\alpha$$:

$$\alpha l_0 = \frac{\Delta l}{\Delta T}$$

where $$l$$ is the length of the system and $$T$$ is it's temperature.

This is the same as saying:

$$\frac{l}{l_0} - 1 = \alpha T$$

One can assume this to mean if there are $$N$$ particles the average spacing in a uniform material's particles is given by:

$$\frac{\langle x_i - x_{i+1} \rangle}{l_0} - \frac{1}{N} = \frac{\alpha T}{N}$$

where $$x_i$$ is the $$i$$'th particle's position adjacent to the $$i+1$$ particle's position and the length of the system is given by $$l \sim \langle x_1 - x_N \rangle$$

Multiplying $$N$$ both sides:

$$N \frac{\langle x_i - x_{i+1} \rangle}{l_0} - 1 = \alpha T$$

where $$\langle x_1 \rangle \geq \langle x_2 \rangle \geq \langle x_3 \rangle \geq \langle x_4 \rangle \geq \dots$$

Let's write everything we have gathered so far in terms of the density matrix:

$$\rho = \sum_{ij} p_{ij} |\psi_i \rangle \langle \psi_j |$$

The temperature is given by:

$$T= \Big( \frac{\partial S}{\partial U} \Big)_{N,V} =\Big(\frac{\partial \text{Tr}(\rho \ln\rho) }{ \partial \text{Tr}( \rho H)}\Big )_{N,V}$$

Where $$H$$ is the Hamiltonian. The average spacing is given by: $$\langle x_i - x_{i+1} \rangle =\text{Tr}(\rho(x_i - x_{i+1}))$$

Thus, we have a differential equation the density matrix must satisfy:

$$\frac{N}{l_0} \text{Tr}(\rho(x_i - x_{i+1})) - 1 = \alpha \Big(\frac{\partial \text{Tr}(\rho \ln\rho) }{ \partial \text{Tr}( \rho H)}\Big )_{N,V}$$

## Question

Does such an approach already exist in the literature? What are some example Hamiltonian and density matrices which obeys this equation? Can a more explicit condition that the density matrix must obey be constructed (perhaps using $$\text{Tr}(A \otimes B) = \text{Tr}(A) \text{Tr}(B)$$)? Can this describe a strongly coupled system (my reason for asking about strongly coupled systems is that liquid mercury is a strongly coupled system)?

## Edit: My attempt

Let us define $$N/l_0 = \kappa$$ as a constant.

$$\kappa \text{Tr}(\rho(x_i - x_{i+1})) - 1 = \alpha \Big(\frac{\partial \text{Tr}(\rho \ln\rho) }{ \partial \text{Tr}( \rho H)}\Big )_{N,V}$$

Multiplying both sides with $${ \partial \text{Tr}( \rho H)}_{N,V}$$:

$$\kappa \text{Tr}(\rho(x_i - x_{i+1}) \partial \text{Tr}( \rho H)_{N,V} - \partial \text{Tr}( \rho H)_{N,V} = \alpha {\partial \text{Tr}(\rho \ln\rho)_{N,V} }$$

Using $$A \partial B = \partial (AB) - B \partial A$$:

$$- \kappa \text{Tr}( \rho H) \partial \text{Tr}(\rho(x_i - x_{i+1}))_{N,V} - \partial \text{Tr}( \rho H)_{N,V} = \alpha \partial \text{Tr}(\rho \ln\rho)_{N,V} - \partial( \kappa \text{Tr}( \rho H) \text{Tr}(\rho(x_i - x_{i+1}))_{N,V} )$$

Multiplying $$- e^{ \kappa \text{Tr}( \rho(x_i - x_{i+1}))}$$ both sides:

$$\partial (e^{ \kappa \text{Tr}( \rho(x_i - x_{i+1}))} \text{Tr}( \rho H) )_{N,V}= (e^{ \kappa \text{Tr}( \rho(x_i - x_{i+1}))})\partial( \kappa \text{Tr}( \rho H) \text{Tr}(\rho(x_i - x_{i+1}))_{N,V} -\alpha ( e^{ \kappa \text{Tr}( \rho(x_i - x_{i+1}))}){\partial \text{Tr}(\rho \ln\rho)_{N,V} }$$

Integrating both sides and using a constant $$c$$:

$$\text{Tr}( \rho H) = -\alpha e^{ -\kappa \text{Tr}( \rho(x_i - x_{i+1}))} \int ( e^{ \kappa \text{Tr}( \rho(x_i - x_{i+1}))}){\partial \text{Tr}(\rho \ln\rho)_{N,V} } + c e^{ \kappa \text{Tr}( \rho(x_i - x_{i+1}))} + e^{-\kappa \text{Tr}( \rho(x_i - x_{i+1}))} \int (e^{ \kappa \text{Tr}( \rho(x_i - x_{i+1}))})\partial( \kappa \text{Tr}( \rho H) \text{Tr}(\rho(x_i - x_{i+1}))_{N,V}$$

Now, we can substitute the L.H.S in the R.H.S and get some sort of series(?)

• Just for clarification: $|\psi_i\rangle$ ist the full, $N$-parrticle, state or one-particle state at any of the sites? Are the single particles confined in 0-d (at any of the positions, $n\in\lbrace1,\ldots,N\rbrace$ which are constant, in some state $i$) or in 1-d (free, possibly interacting particles confined onsome interval with given one-particle states)? May the particles jump between sites? If so: bosons or fermions and what are onsite interactions? – denklo Nov 12 '18 at 8:24
• Physically I was thinking of a $1$ dimensional thermometer (in some sense) ... So I was thinking of a Hamiltonian like $\hat H = p_1^2/2m + p_2^2 / 2m + \dots + V(x_1,x_2,x_3,\dots)$ where $V$ is the potential. Hence, $|\psi \rangle$ would be the full $N$ particle state .. I was thinking more along the lines of fermions .. – More Anonymous Nov 12 '18 at 8:55