Our empirical equation is (quoted from Wikipedia https://en.wikipedia.org/wiki/Proton)

G m p r p × 1   kg = 3 k T (1)

G: gravitational constant, 6.6743 × 10⁻¹¹ (m³∙kg⁻¹∙s⁻²).

m_p: the rest mass of a proton, 1.67 × 10⁻²⁷ (kg).

r_p: charge radius, 8.41 × 10⁻¹⁶ (m).

k: Boltzmann constant, 1.38 × 10⁻²³ (J/K).

T: temperature (K).

1 kg: the standard mass (kg).

The temperature (T) calculated using this formula was 3.20 K, which is similar to the temperature of the cosmic microwave background of 2.73 K. The standard mass (1 kg) must be used due to the definition of the gravitational constant. This problem was explained in Section 3.1.

Equations (2)-(4) are our conclusions.

ϕ = 3 k T . (2)

m p c 2 = h ν − 3 k T (3)

d S Gibbs d r T = 3 n k T r (4)

where h and v are the Planck constant and the frequency, respectively. n is the number of protons, and r is the distance from a large mass (n × m_p).

This section is strongly influenced by Yasuo Katayama [5]. http://home.catv.ne.jp/dd/pub/pot/pot.html#0.

He thought that, according to Newton, the energy can be obtained infinitely from the gravitational potential energy, but according to Einstein, there should be an upper limit (mc²). Thus, the rest mass should decrease with decreasing the gravitational potential. Equation (16) was quoted from Yasuo Katayama.

The gravitational potential (Φ(r)) is given by

Φ ( r ) = − G M r , (5)

where M and r are the mass and the distance from the mass, respectively. Then, the work (W) is given by

W = m × ( Φ ( r 2 ) − Φ ( r 1 ) ) = m × ( G M r 1 − G M r 2 ) (6)

where m is a sufficiently small mass. r₁ and r₂ (>r₁) are the distances from mass M. Therefore,

W = ∫ r 1 r 2 m × d Φ ( r ) d r d r (7)

Then,

d W d r = m g (8)

g = d Φ ( r ) d r (9)

Next, from the mass-energy equivalence and the Law of the Conservation of Energy, we obtain

m 2 c 2 = m 1 c 2 + W (10)

Therefore,

m 2 − m 1 = W c 2 . (11)

m should increase with the increasing of r.

d m d r = 1 c 2 d W d r . (12)

From Equations (5) and (9), we obtain

d m d r = 1 c 2 m g . (13)

Therefore, from Equation (6) and Equation (10), we find that

d m d r = 1 c 2 m d Φ ( r ) d r . (14)

Then,

d m = 1 c 2 m d Φ ( r ) . (15)

Then, we assumed that m should be the minimum mass (m_p1) when r is the minimum distance (r_p1).

ln ( m p 2 m p 1 ) = 1 c 2 ( Φ ( r p 2 ) − Φ ( r p 1 ) ) (16)

Thus,

ln ( m p 2 m p 1 ) = G c 2 ( M 1 r p 1 − M 2 r p 2 ) (17)

When M is n (the number of protons) × m_p,

M 1 = n × m p 1 (18)

M 2 = n × m p 2 (19)

From Equations (17)-(19),

ln ( m p 2 m p 1 ) = n G c 2 × ( m p 1 r p 1 − m p 2 r p 2 ) (20)

when r_p2 is infinitely large.

ln ( m p 2 m p 1 ) = n G c 2 × m p 1 r p 1 (21)

Therefore,

c 2 n ln ( m p 2 m p 1 ) = G × m p 1 r p 1 (22)

When M₁ is 1 kg, n is 6.02 × 10²⁶ (=1000 × 10²³). The value of n is calculated from Avogadro’s number.

1   kg = n × m p 1 (23)

thus forming Equation (22) and Equation (23),

m p 1 c 2 × ln ( m p 2 m p 1 ) = 1   kg × G × m p 1 r p 1 (24)

Then, n disappeared. This does not mean that n is unimportant. As shown in Equation (20), n is clearly meaningful. We tried to explain the influence of 1 proton on the gravitational potential, and then, n disappeared. When Newton defined 10 kg as the standard mass, the value of G should change. However, Equation (24) should not change. The real meaning of Equation (24) was discussed in the later section.

This section is the most important and is quoted from our own experimental results [4]. In this section, we tried to explain the concept of the solvation Hamiltonian mean force from the perspective of thermodynamics in relation to hopping conduction.

A solid oxide fuel cell (SOFC) directly converts the chemical energy of a fuel gas, such as hydrogen or methane, to electrical energy. A solid oxide film is used as the electrolyte, where the main carriers are oxygen ions. As electrolytes, YSZ and SDC (samaria-doped ceria) are well known. The open circuit voltage (OCV) is calculated from Wagner’s equation.

J O 2 = − R T 16 F 2 L ∫ ln ( p O 2 anode ) ln ( p O 2 cathode ) σ el σ ion σ el + σ ion d ln ( p O 2 ) (25)

where J O 2 and p O 2 are the O₂ flux and the O₂ partial pressure, respectively; p O 2 cathode and p O 2 anode are the O₂ partial pressures at the cathode and the anode, respectively; R, T, and F are the gas constant, the absolute temperature, and Faraday’s constant, respectively; L is the thickness of the membrane or film; and σ_el and σ_ion are the conductivities of the electrons and oxygen vacancies, respectively.

However, we have previously reported two experimental problems related to Wagner’s equation.

Problem 1: The unchanged OCV during electrode degradation [6] [7].

Problem 2: The equilibration process in response to a change in the anode gas using thick SDC electrolytes [8].

To solve these problems, we proposed the concept of the solvation Hamiltonian of mean force (ϕ), which was discussed by Jarzynski theoretically [3]. After introducing the concept of the solvation Hamiltonian of mean force (ϕ), we noticed that our model was still compatible with Wagner’s equation. Thus, we called our model “the modified Wagner’s equation”.

Using only three graphs, the solvation Hamiltonian of mean force (ϕ) can be explained. The Boltzmann distribution of oxygen ions in the electrolyte at 1073 K is displayed in Figure 2. The ions with energies exceeding the ionic activation energy become carriers (hopping ions). Figure 3 presents an incorrect carrier distribution. According to Jarzynski, an accurate distribution should be a canonical ensemble, as shown in Figure 4. The solvation Hamiltonian of mean force (ϕ) can be defined and is illustrated in Figure 3.

Ions with energies exceeding the ionic activation energy are converted into charge carriers (i.e., hopping ions).

According to Jarzynski, an accurate distribution should be a canonical ensemble, and the solvation Hamiltonian of mean force (ϕ) can be defined [3].

The shape of the distribution in this figure should be the same as the shape of the distribution in Figure 1.

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

η i = μ i + z i F φ (26)

where z_i, η_i, μ_i and φ are the valence of species i, the electrochemical potential, the chemical potential and the electrostatic potential, respectively. In Equation (26), the following transformation should be considered during ion hopping:

μ i _hopping = μ i _vacancies + N E a (27)

z i F φ hopping = z i F φ vacancies − N E a (28)

where N, μ_{i_hopping}, μ_{i_vacancies}, φ_{i_hopping}, and φ_{i_vacancies} are Avogadro’s number, the chemical potential of hopping ions, the chemical potential of ions in vacancies, the electrical potential of hopping ions, and the electrical potential of ions in vacancies, respectively. For example, E_a is 0.7 eV in SDC electrolytes,

φ hopping = φ vacancies + E a 2 e = φ vacancies + 0.35   V (29)

When there are enough electrons from the anode side in the SDC electrolyte, due to microscopic interactions, this voltage (0.35 V) should be neutralized, and the electrical potential energy of ions should be dissipated by electrons. When the Nernst voltage at 1073 K is 1.15 V, the measured open circuit voltage (OCV) at 1073 K should be 0.80 V (=1.15 V − 0.35 V). Strictly speaking, the voltage loss can be calculated as follows,

OCV loss = E a 2 e × ( 1 − t ion ) (30)

t ion = σ ion σ el + σ ion (31)

where OCV_loss and t_ion are the voltage loss in the OCV and the ionic transference number, respectively. We conclude that the solvation Hamiltonian of mean force (ϕ) should be

ϕ = N E a . (32)

Therefore, from Equation (27) and Equation (28),

μ i _hopping = μ i _vacancies + ϕ (33)

z i F φ hopping = z i F φ vacancies − ϕ (34)

From Equation (30),

Δ E s = ϕ × ( 1 − t ion ) (35)

where ΔE_s is the fixed entropy energy difference. Equation (33), Equation (34) and Equation (35) demonstrate the central concept of “the modified Wagner’s equation”.

The compatibility between the Jarzynski’s equality and the modified Wagner’s equation is discussed in this section. According to Jarzynski [3], the Gibbs volume entropy is given by

W ≥ 3 2 N k T [ ( V A V B ) 2 3 − 1 ] ≥ N k T ln ( V A V B ) . (36)

In the area of solid-state ionics, the Nernst equation is

OCV th = N k T 4 F ln ( p O 2 A p O 2 B ) , (37)

where OCV_th is the Nernst voltage. Similar to Equation (36) and Equation (37), the study on the OCV in the solid-state ionics is related with the Gibbs volume entropy.

The nonequilibrium work relation is described by Jarzynski’s equality [3].

〈 e − β W 〉 = e − β Δ F T (38)

where β, W and ΔF_T are 1/kT, the work and the difference in the Helmholtz free energy, respectively. k is the Boltzmann constant. The angular brackets <-> indicate an ensemble over the realizations of the process.

For adiabatic processes, ΔF_T is zero. The following equation should be added [3]:

〈 e − β X 〉 = 1 (39)

where X is the energy for the same adiabatic process and ensemble of realizations. In the area of solid-state ionics, X is the ionic activation energy. From Equation (39), an accurate distribution of hopping ions in the electrolytes should be a canonical ensemble. Thus, the Boltzmann distribution during the ion hopping shown in Figure 4 can be explained by the Jarzynski’s equality.

Furthermore, the solvation Hamiltonian of mean force (ϕ) and the fixed entropy energy difference (ΔE_s) in Equation (39) can be defined mathematically [3]. Their mathematical approach is too general. In the area of the generalized second law, the separation of the Boltzmann distribution is called “Maxwell’s demon”.

Consequently, the concept of “the modified Wagner’s equation” can be the specialized version of the Jarzynski’s equality.

Ted Jacobson discovered that the general relative theory is governed by the following Equation (57).

d S = ∂ S ∂ E d E + ∂ S ∂ V d V (57)

where S, E and V are the entropy, the internal energy and the volume, respectively.

Equation (46) is rewritten.

m p c 2 = h ν − 3 k T (58)

In Equation (58), internal energy (E) should decrease. However, the entropy should not decrease. From Equation (57), the Gibbs volume entropy (S_Gibbs) can be defined.

S Gibbs ( m p c 2 ) × T = S Gibbs ( h ν ) × T + 3 k T (59)

Then, we must consider the degree of freedom (x, y and z direction). In Equation (59), the Gibbs volume entropy (S_Gibbs) for one direction is

S Gibbs ( m p c 2 ) = S Gibbs ( h ν ) + k T (60)

Next, dV in the distance (r) from the mass should be

d V = 4 π r 2 × d r (60)

Thus,

∂ S Gibbs ∂ V d V = S Gibbs 4 3 π r 3 × 4 π r 2 × d r = 3 S Gibbs r d r (61)

Therefore, from Equation (60) and Equation (61),

d S Gibbs d r T = 3 k T r (62)

When the number of protons is n,

d S Gibbs d r T = 3 n k T r (63)

It is important that the direction is opposite to that of the gravitational potential. However, Equation (63) is similar to Newton’s Law. k is one of the “defining constants” and becomes part of the 2019 redefinition of SI base units. However, from Equation (53), the value of k should be changed. Equation (53) is rewritten.

3 k 1 T m p 1 c 2 = 8.8 × 10 − 13 (64)

It is possible that the increase in the rest mass can be observed on the galaxy scale. Then, k should increase with increasing the mass of a proton. This may provide a solution for the dark matter problem.

The solvation Hamiltonian mean force (ϕ) is assumed to originate from the asymmetry of space-time. The asymmetry of space-time was already discovered and is known as “chirality”. According to Einstein,

x 2 + y 2 + z 2 − c t 2 = 0 (65)

We proposed the following equation.

Z 1 2 + Z 2 2 + Z 3 2 + Z 0 2 = 0 (65)

where Z₁, Z₂, Z₃ and Z₀ are complex numbers. Then, the asymmetry e^iθ can be defined. In the area of solid-state ionics, the asymmetry can be produced artificially. For example, YSZ (yttria stabilized zirconia) can be obtained from ZrO₂ with an addition of 8% Y₂O₃. Then, ions can move in the YSZ electrolytes due to the existence of vacancies. Therefore, the ionic activation energy (E_a) can be defined for the ionic conduction. We noticed that the ionic activation energy can be defined as the solvation Hamiltonian mean force (ϕ). Then, the Gibbs volume entropy can be measured experimentally. Consequently, we proposed the fifth dimension, which is the internal dimension originating from the asymmetry that acts to separate the Boltzmann distribution.