## 3 Results  
In this paper, we have developed a non-equilibrium thermodynamic analysis of the cell membrane electric potential in order to explain, analytically, the role of the ions fluxes in relation to cancer behaviour, but also to the thermoelectric properties of the membrane itself. Indeed, the Equation (7) allows us to state that an increase in the cell temperature (development of any inflammation) implies a decrease in the membrane electric potential. Always in relation to Equation (7), we can highlight that the values of \(\alpha\), \(T\) and \(T_{0}\) are approximately the same for cancer and normal cells; therefore, from a physical viewpoint, the difference between cancer and normal cells must be expressed in terms of the thermoelectric coefficient \(k\). In particular, experimental results point out that the membrane of cancer cells is depolarized [1,12], consequently, in relation to our Equation (7), the thermoelectric coefficient for cancer results greater than the one of a normal cell. Moreover, during its life cycle, cell membranes have continue transitions between Seebeck-like and Peltier-like effects to sustain heat and mass fluxes. These continuous transitions are the thermo-electric cycle responsible for respiration, metabolism, reorganisation, proliferation, communication and all the biophysical and biochemical processes inside the cells [39]. During the Seebeck-like effect, the membrane exchanges heat towards the environment, with a related decrease in its entropy, in accordance with the Schrodinger approach. During the Peltier-like effect, the membrane exchanges ions, metabolites and waste molecules with the environment in order to realise the biochemical processes for life and proliferation. Here, we develop a numerical evaluation in relation to the Ca\({}^{2+}\)-flux. This example is very important due to the fundamental role played by the Ca\({}^{2+}\) ion in the regulation of a great number of cell functions [50; 51; 52; 53; 54]. To do so, we write the Equation (11) as follows:  
\[T\,\frac{ds}{dt}=-\
abla\cdot\left(\mathbf{J}_{u}-\mu_{\text{Ca}}\,\mathbf{J} _{\text{Ca}}\right) \tag{18}\]  
This last equation, considering \(T\) constant, and following Prigogine (\(ds/dt=0\)) [55], becomes  
\[\
abla\cdot\left(\mathbf{J}_{u}-\mu_{\text{Ca}}\,\mathbf{J}_{\text{Ca}}\right)=0 \tag{19}\]  
Now, we introduce the First Law of Thermodynamics for the cell membrane, so we can write [50]  
\[\frac{du}{dt}\,dV=\rho\,c\,\frac{dT}{dt}\,dV=\delta\dot{Q}=-\alpha\left(T-T_{0} \right)dA \tag{20}\]  
where \(\rho\approx 10^{3}\) kg m\({}^{-3}\) is the cell density, \(c\approx 4186\) J kg\({}^{-1}\) K\({}^{-1}\) is the specific heat of the cell, \(\alpha\approx 0.023 Re^{0.8}Pr^{0.35}\lambda/\langle R\rangle\) is the coefficient of convection, with \(\lambda\approx 0.6\) W m\({}^{-1}\)K\({}^{-1}\) conductivity, \(Re\approx 0.2\) the Reynolds number and \(Pr\approx 0.7\) the Prandtl number [43], \(A\) area of the cell membrane, \(V\) is the cell volume and \(\beta=\alpha\,dA/dV\) is constant. Now, considering Equation (10), we can write:  
\[\
abla\cdot\mathbf{J}_{u}=\alpha\,\frac{dA}{dV}\left(T-T_{0}\right) \tag{21}\]  
and, it follows [50]:  
\[\
abla\cdot\left(\mu_{\text{Ca}}\mathbf{J}_{\text{Ca}}\right)=\frac{\delta Q} {dV}=\alpha\,\frac{dA}{dV}\left(T-T_{0}\right) \tag{22}\]  
\[J_{\text{Ca}}=\frac{\ell\cdot\pi}{\mu_{\text{Ca}}\langle R\rangle}\left(T-T_{0 }\right)=\frac{4\times 10^{-9}\times 0.023\times 0.02\times 0.0 ^{70.35}\times 0.6\times 0.4}{-552.79\times 10^{3}\cdot(R)^{2}}=\frac{-0.97\times 10^{- 17}}{\langle R\rangle^{2}}\text{ [mol s${}^{-1}$m${}^{-2}$]} \tag{23}\]  
where \(\ell\approx 0.004\) um is the depth of the cell membrane [56] and \(\langle R\rangle\) is the mean radius of the cell, considered, in the first approximation, as a sphere, \(\mu_{\text{Ca}}=-552.79\) kJ mol\({}^{-1}\) and \(T-T_{0}\approx 0.4\)\({}^{\circ}\)C [57]. The numerical result depends on the mean size of the cell.