_{1}

Bell’s non-locality theorem can be understood in terms of classical thermodynamics, which is already considered to be a complete field. However, inconsistencies in classical thermodynamics have been discovered in the area of solid-oxide fuel cells (SOFCs). The use of samarium-doped ceria electrolytes in SOFCs lowers the open-circuit voltage (OCV) to less than the Nernst voltage. This low OCV has been explained by Wagner’s equation, which is based on chemical equilibrium theory. However, Wagner’s equation is insufficient to explain the low OCV, which should be explained by fluctuation and dissipation theorems. Considering the separation of the Boltzmann distribution and Maxwell’s demon, only carrier species with sufficient energy to overcome the activation energy can contribute to current conduction, as determined by incorporating different constants into the definitions of the chemical and electrical potentials. Then, an energy loss equal to the activation energy will occur because of the interactions between ions and electrons. This energy loss means that an additional thermodynamic law based on an advanced model of Maxwell’s demon is needed. In this report, the zero-point energy can be explained by this additional ther-modynamic law, as can Bell’s non-locality theorem.

Solid-oxide fuel cells (SOFCs) directly convert the chemical energy of fuel gases, such as hydrogen and methane, into electrical energy. SOFCs use a solid-oxide film as the electrolyte, and oxygen ions serve as the main charge carriers. Typically, yttria-stabilized zirconia (YSZ) is used as the electrolyte material in these cells. The operating temperature of these cells (873 - 1273 K) should be reduced to extend their life spans. Therefore, the use of mixed ionic and electronic conducting (MIEC) electrolyte materials, such as samaria-doped ceria (SDC) electrolytes, at lower temperatures is preferred. The open- circuit voltage (OCV) of an SDC cell is approximately 0.8 V, which is lower than the Nernst voltage (V_{th}) of 1.15 V at 1073 K. This low OCV is attributed to the low value of the ionic transference number (t_{ion}). The low OCV has been calculated using Wagner’s equation, which is based on chemical equilibrium theory [

Over the past two decades, the understanding of nonequilibrium thermodynamics has been enhanced by fluctuation and dissipation theorems such as the Jarzynski and Crooks relations [

where

Then, the inevitable energy dissipation equal to

Bell’s theorem [

The low OCV has been explained by Wagner’s equation, which is based on chemical equilibrium theory. The flux of oxygen ions under open-circuit conditions, as described by Wagner’s equation [

where

From Equation (2), Equations (3) and (4) can be deduced [

where R_{i} and I_{i} are the ionic resistances of the electrolyte and the ionic current, respectively.

Equation (4) is often used when _{i} is not apparently expressed. When an SOFC is operated under OCV conditions, the electronic leakage current (I_{e}) is defined using the external current (I_{ext}):

In general, I_{i} and I_{e} are not zero. Parameter t_{ion} is expressed as

However, σ_{el} is a function of the

where

Problem 1: The calculated OCV will always be higher than the experimental result. Polarization voltage losses are frequently used to compensate, but the lack of change of the OCV during electrode degradation is impossible to explain [

Problem 2: The equilibration process of thick SDC electrolytes in response to a change in the anode gas cannot be explained because the response from the experimental results is too fast to explain the delay in the electron diffusion current from the cathode to the anode [

To solve the aforementioned problems, there should be a fast response current- independent constant anode voltage loss (_{i}I_{i} becomes very small because of the anode shielding effect [

YSZ is a ceramic in which the crystal structure of zirconium dioxide is made stable by the addition of yttrium oxide. When yttria is added to pure zirconia, Y^{3+} ions replace Zr^{4+} on the cationic sublattice. Oxygen vacancies are generated due to the charge neutrality. A graphical explanation is provided in

hopping from an occupied site (8c site) to a vacant lattice site (8c). Oxygen ions then pass through the first gate (32f site) and the second gate (another 32f site) [

YSZ electrolytes are purely ionic conductors; however, SDC electrolytes are MIEC materials. Interactions should occur between hopping ions and electrons. A schematic of the path for ions in SDC electrolytes is shown in

In previous work, we explained the advanced model of Maxwell’s demon electrochemically [_{real}) should be determined using Equation (8) [

Then, voltage losses occur during every hopping process in the MIEC area. Therefore, the voltage loss resulting from only four hopping processes (

The Jarzynski equality is [

where the over-line indicates an average over all possible realizations of an external process that takes the system from the equilibrium state A to a new nonequilibrium state under the same external conditions as that of the equilibrium state B.

Ions in equilibrium state A lose energy (=0.7 eV) during their hopping in the MIEC electrolyte, and the heating of the colder ions can occur immediately in the vacancies. However, after the last hop, the ions cannot be heated and therefore exit the electrolyte in equilibrium state B. A schematic of this explanation of the energy loss is shown in

The electrochemical potential can be separated into the chemical potential and the electrostatic potential,

where z_{i}, η_{i}, μ_{i} and

where N, _{a} is 0.7 eV,

The electrical potential (

“Considering the separation of the Boltzmann distribution and Maxwell’s demon, only carrier species with sufficient energy to overcome the activation energy can contribute to current conduction, as determined by incorporating different constants into the definitions of the chemical and electrical potentials. Then, an energy loss equal to the activation energy will occur because of the interactions between ions and electrons.”

Using S (the action function), we can state the Nernst-Planck equation as

From the principle of least action,

Considering the separation of the Boltzmann distribution and Maxwell’s demon, the generalized expression obtained from Equations (12) and Equation (13) is

Equation (17) holds for one hop but not zero when E_{a} is not zero. In SDC electrolytes, the energy loss during hopping due to inevitable dissipation (E_{loss}) is

Oxygen ions transport through a path during hopping in the lattice structure. The electric potential in the path during hopping differs from the electric potential in the vacancies. In YSZ electrolytes, which are purely ionic conductors, ions do not lose energy. However, in SDC electrolytes, which are MIEC materials, the ions lose a large amount of energy. However, measuring the electric potential during hopping is difficult. We thus propose a determination experiment. A negatively charged metal ball (N-ball) and positively charged heavy metal balls (P-balls) should all be coated with plastic. Using springs, the three P-balls can be connected with each other, and then the N-ball can go through the center of them at high speed under vacuum. The speed of the N-ball may be unchanged after the collision. However, when an ionized gas spray is used during the collision, the speed of the N-ball will become small. The energy loss of the N-ball can be controlled by changing the spring constant. A schematic of the explanation of this determination experiment is shown in

Equation (17) and Equation (18) form the generalized expression of the additional thermodynamic law. In quantum physics, instead of_{a}, K.E. (kinetic energy), P.E. (potential energy) and

where _{0}) for one dimension is [

Equation (20) indicates that, when the separation of the Boltzmann distribution by

Bell’s theorem is a “no-go theorem” that draws a distinction between quantum mechanics and classical mechanics [

The inequality that Bell derived can then be written as

where

proven by considering a system of two entangled qubits. The examples concern systems of particles that are entangled in spin. Quantum mechanics allows predictions of correlations that would be observed if these two particles have their spin measured in different directions.

However, Bell was unaware of the inevitable energy dissipation during the measurement. According to Brillouin [

In statistics, a confounding factor is an extraneous factor in a statistical model that correlates with both the dependent variable and the independent variable in a way that explains away some of the correlation between these two variables.

When there should be inevitable energy dissipation from particles to the environment during the measurement, combined with the relative theory, the total angular momentum of two entangled qubits cannot be conserved after the measurement. We propose the following confounding factor based on locality.

Confounding factor: Spin is similar to a vector quantity that has a direction as the local hidden variable. However, the information of the direction can be erased because of the inevitable energy dissipation during the measurement. This information refers to spin in the positive or negative direction of the chosen axis after the measurement.

We consider a photon. The frequency of photons is much higher than that of radio waves. Within radio waves, AM broadcasting is the process of radio broadcasting using amplitude modulation and FM broadcasting uses frequency modulation. In principle, we can manipulate photons using amplitude modulation and frequency modulation. When the angle between a vector quantity and the chosen axis is_{0}), deleting the amplitude modulation and frequency modulation, can be defined using the following equation:

Then, the rest of the information I_{1} between

Thus, I_{1} is determined only by

In general, because of rotational symmetry, Equation (24) should be rewritten as

In the experiment performed by Aspect [

Consequently, Bell’s non-locality theorem can be understood in terms of classical thermodynamics.

We proposed a determination experiment for local hidden variables, which is combined with the experiment of Aspect and a double-slit experiment. The source S produces pairs of photons, which are sent in opposite directions. After the detection of the left photons, the right photon receives the information of its position by quantum teleportation. Then, an interference pattern cannot be created on the screen behind the double slit. A schematic of the determination experiment for the local hidden variables is shown in

Inconsistencies in classical thermodynamics have been discovered in the area of SOFCs. The use of SDC electrolytes in SOFCs lowers the OCV to less than the Nernst voltage. This low OCV has been explained by Wagner’s equation, which is based on chemical equilibrium theory. However, Wagner’s equation is insufficient to explain the low OCV. By considering the separation of the Boltzmann distribution and Maxwell’s demon, we proposed an additional thermodynamic law based on an advanced model of Maxwell’s demon. The zero-point energy can be explained by this additional thermodynamic law. Then, inevitable energy dissipation equal to

Maxwell’s demon. Bell’s non-locality theorem can be explained by this additional thermodynamic law. Spin is similar to a vector quantity with direction as a local hidden variable. According to Brillouin, the information of the direction can be erased because of the inevitable energy dissipation during the measurement. We proposed a determination experiment for local hidden variables, which was combined with the experiment by aspect and a double-slit experiment. When an interference pattern is observed, quantum teleportation (non-locality) will be denied. Bell’s non-locality theorem can be local with this additional thermodynamic law.

Miyashita, T. (2017) Bell’s Non-Locality Theorem Can Be Understood in Terms of Classical Thermodynamics. Journal of Modern Physics, 8, 87-98. http://dx.doi.org/10.4236/jmp.2017.81008