### Variation of NIR spectra

The variation in the NIR spectra with respect to the *EY* is shown in Fig. 1a. As the original spectra did not show clear tendencies with *EY*, we processed them as a second derivative treatment. As a result, the absorption intensity exhibited characteristic changes in specific bands. For instance, a gradual decrease in the absorption over time was observed in the peaks at 5848 cm^{−1}, which were assigned to the first overtone of the C–H stretching mode of furanose/pyranose owing to hemicellulose [23]. This result was consistent with those of previous reports about the age dependency of hemicellulose content [11, 24].

The variation pattern of wood components with *EY* was not so obvious, although the second derivative spectra detected a successive decrease in hemicellulose. Moreover, it is insufficient to consider the changes in each variable individually as changes in one property induce changes in other properties. Hence, we did not consider the spectral variation as an individual variable (i.e., wavenumber), but rather as a whole for the physical system with multiple degrees of freedom [16]. This idea is closely related to statistical mechanics. The basic idea of statistical mechanics is that instead of considering a single dynamic system, we considered a collection of systems is considered that correspond to the same Hamiltonian, that is, a statistical ensemble [25, 26]. Herein, we assumed that the spectral variables were microscopic elements comprising the wood system. In other words, the NIR spectral matrix from an ensemble of wood samples could be identified with a point cloud in the *n*-dimensional phase space [27]. Thus, spectral variables serve as generalized coordinates in analytical mechanics [28, 29].

### Changes of the eigenvalue distribution

As the NIR spectral matrix represents the variation in the ensemble of wood samples, the distribution of the eigenvalues calculated from the matrix should provide inclusive information of the ensemble. Moreover, the characteristic equation to solve the eigenvalue problem is invariant with respect to changes in the basis [30]. Therefore, the set of eigenvalues reveals the intrinsic variation of the wood ensemble. The distributions of eigenvalues (*E*_{i}) for representative *EY* are shown in Fig. 2. The eigenvalues in older *EY* were widely distributed compared to those in younger *EY*, indicating that the NIR spectral matrix varied in a more orderly manner as the *EY* increased. This is consistent with a previous study in which eigenvalues were widely distributed with cambial age [15].

As mentioned above, the set of eigenvalues represents the energy state of the system in the context of physics [20, 31, 32]. Therefore, changes in wood over time can be analyzed from the perspective of thermodynamics and statistical mechanics [21, 33]. Kabashima and Takahashi (2012) showed that the first eigenvalue *E*_{1} of the variance–covariance matrix can be evaluated using the Helmholtz-free energy (*F*) in the limits of *β* → ∞ [34]:

$${E}_{1}=2\underset{\beta \to \infty }{\mathrm{lim}}F\left(\beta |{\varvec{C}}\right),$$

(5)

where *F* is defined as a function of *β* conditioned by the matrix *C*, that is, \(F\left(\beta |{\varvec{C}}\right)=-{\beta }^{-1}\mathrm{log}Z\left(\beta |{\varvec{C}}\right)\). Figure 3a shows the trajectory of some representative eigenvalues with respect to *EY*. Consistent with the results shown in Fig. 2, the eigenvalues diffused without intersecting with each other. A clear increase in the first eigenvalue, which is equivalent to the Helmholtz-free energy, suggests that the xylem in heartwood changes to a more ordered physical state with an increase in *EY*. Based on the aforementioned suggestion, we confirmed the equivalence between the xylem in the heartwood of the standing tree and the archaeological wood. The superior mechanical properties and dimensional stability of archaeological wood are largely due to changes in the crystallinity of cellulose, that is, the regularity of its molecular configuration. Although many other properties also vary simultaneously with the mechanical properties and changes in dimensional stability, the diffusion of eigenvalues indicates that these complex variations represent the progress of the organization of the xylem tissue. As the diffusion of eigenvalues was also observed with an increase in *CA* [15], the phenomena should progress in both physiological and anti-physiological aging processes.

An *n* × *n* Hermitian matrix has *n* eigenvalues that move along the real axis, and the stochastic process of the eigenvalues is called Dyson’s Brownian motion [35]. Although the variance–covariance matrix is a real-symmetric, we discuss the time evolution of the eigenvalue distribution with respect to *EY* using the Dyson model. The stochastic dynamics of the eigenvalues {*E*_{1}, *E*_{2}, …, *E*_{n}} of the matrix are described as follows:

$$dE_{i} = dB_{i} + \sum\limits_{\begin{subarray}{l} 1 \le j \le n \\ j \ne i \end{subarray} } {\frac{dt}{{E_{i} - E_{j} }}} ,\,\,\,\,\,\,\,1 \le i \le n,$$

(6)

where *B*_{i} (1 ≤ *i* ≤ *n*) represents independent one-dimensional Brownian motions [33, 35]. From Eq. (6), the probability density of the eigenvalues, \(p=p({E}_{1},\cdots ,{E}_{n},t)\), satisfies the Fokker–Planck equation [36]:

$$\frac{\partial }{\partial t}p = \sum\limits_{i = 1}^{n} {\sum\limits_{j \ne 1} {\frac{1}{{E_{i} - E_{j} }}} } \frac{\partial p}{{\partial E_{i} }} + \frac{1}{2}\sum\limits_{i = 1}^{n} {\frac{{\partial^{2} p}}{{\partial E_{i}^{2} }}}$$

(7)

The trajectory of the differential of *p* with respect to the *EY* was calculated using Eq. (7), as shown in Fig. 3b. The differential coefficient of *p* increased negatively with *EY*. It remains unclear why the opposite result was found in the case of cambial aging [15]. However, the complicated aging process can be understood if we solve the stochastic differential equation (Eq. 7) representing the motion of the eigenvalues, which contains all the variations generated by various wood properties. Moreover, it is important to note that the current results did not represent the behavior of individual trees, but that of a statistical ensemble with a large number of elements [20]. Hence, the trajectory of the Fokker–Planck equation should exhibit highly universal behavior based on the law of large numbers [37].

### Randomness and irreversibility of the aging process

Here, we considered the changes in wood properties with respect to the elapsed time from the viewpoint of the randomness of the physical system. Using the distribution function *Z* (Eq. 4) as a normalization factor, the probability that the system of interest could occupy the energy eigenstate *E*_{i} can be expressed as

$$p_{i}^{{{\text{can}}}} = \frac{1}{Z}{\text{exp(}} - \beta E_{i} {)}$$

(8)

\({p}_{i}^{\text{can}}\) is equivalent to the canonical distribution in statistical physics [25]. Let *β* be a constant. The distribution of \({p}_{i}^{\text{can}}\) corresponding to each energy eigenstate in the representative *EY* is shown in Fig. 4a. The \({p}_{i}^{\text{can}}\) of the younger *EY* had flatter distributions compared with those of the older *EY*, indicating that the probability distribution of the eigenstate of energy is biased with an increase in *EY*. We can then calculate the Shannon entropy (*S*) as a function of \({p}_{i}^{\text{can}}\) [21]:

$$S = - \mathop \sum \limits_{i = 1}^{N} p_{i}^{{{\text{can}}}} \log p_{i}^{{{\text{can}}}}$$

(9)

Both *Z* and \({p}_{i}^{\text{can}}\) depend on the inverse temperature *β* [see Eqs. (4) and (8)]. The entropy *S* can be evaluated for each *β*. Figure 4b shows the change in entropy *EY* for various *β* conditions. The curves of *S* changed from red to yellow (from top to bottom) as the inverse temperature increased. The domain of *β* was set from 1 to 100,000 to enable entropy calculation. The entropy gradually decreased with *EY* at almost every inverse temperature. These results are consistent with the trend of the Helmholtz-free energy, where the xylem at younger *EY* is in a random state as a physical system, and that at older *EY* is in a more ordered state.

Suppose that the motion of a wood system of m points in *n*-dimensional space is expressed by Hamilton’s equations:

$$\frac{{dq_{i} }}{dt} = \frac{\partial H}{{\partial p_{i} }}, \frac{{dp_{i} }}{dt} = \frac{\partial H}{{\partial q_{i} }},$$

(10)

where \({\varvec{q}}=\left({q}_{1},\cdots ,{q}_{n}\right)\) represents the generalised coordinates and \({\varvec{p}}=\left({p}_{1},\cdots ,{p}_{n}\right)\) represents the momenta. A space of 2*n* dimensions spanned by *q*_{1}, …, *q*_{n}, *p*_{1}, …, *p*_{n} is called the phase space. The point cloud, i.e., an ensemble of wood samples, can be described as a continuous fluid with a density of \(\rho \left({q}_{1},\cdots ,{q}_{n},{p}_{1},\cdots ,{p}_{n},t\right)\) in the phase space [29]. Based on the quantum mechanics [32], the density matrix can be calculated as

$$\rho = \mathop \sum \limits_{i = 1}^{n} {\varvec{u}}_{i}^{{\text{T}}} p_{i}^{{{\text{can}}}} {\varvec{u}}_{i} ,$$

(11)

where *u*_{i} (*i* = 1, …, *n*) is the eigenvector corresponding to eigenvalue *E*_{i} (see Eq. 3). The time evolution of the density matrix is given by the Liouville–von Neumann equation [25]:

$$i\hbar \frac{\partial \rho }{{\partial t}} = \left[ {\rho ,{ }H} \right],$$

(12)

where \(i=\sqrt{-1}\) is imaginary unit, \(\hslash\) is reduced Planck constant, *H* is the diagonal matrix whose elements are eigenvalues calculated using Eq. (3), and the bracket [,] is the commutator defined by two operators [*A*, *B*] = *AB*–*BA*. This can be interpreted as a volume change of the fluid in the phase space. Liouville's theorem states that the fluid volume in the phase space is invariant if the system satisfies the law of conservation [29]. Thus, the density matrix *ρ* is the first integral of Hamilton’s Eq. (10), which is determined by the Hamiltonian *H* (Eq. 3) if, and only if [*ρ*, *H*] = 0. The right-hand side of Eq. (12) was calculated for each *EY* and was a non-zero matrix. The matrix norms of commutator [*ρ*, *H*] exhibited the same trend as the first eigenvalue (data not shown). The density matrix changed significantly with *EY* and did not conserve its volume. As energy invariance refers to the symmetry of a process from a geometrical perspective (Noether’s theorem [25]), the results infer that the aging phenomenon with respect to the elapsed years is an irreversible process, and symmetry breakage during the process becomes more pronounced with the elapsed years. A previous study found similar irreversibility in the aging phenomenon with respect to the *CA* and proposed that wood with the same properties is never renewable [15]. The current study supports this proposition as wood cannot be restored to its original state through the anti-physiological aging process.