Introduction

Thermal noise, also referred to as thermal fluctuation, is ubiquitous in nature. This noise is significant in almost all physical-biochemical processes, including the movement of molecular motors^1-5, adsorption of molecules^6,7, molecular transport ^8-11, and water evaporation^12,13. Thermal noise plays a key role in microscopic measurements^14-16. For example, it can be used to measure the spring constant of atomic force microscope cantilevers¹⁷. Furthermore, the noises associated with biological processes are important when performing single-particle tracking¹⁸. In traditional theories, thermal noise is modeled as Gaussian white noise with a constant power spectrum density (PSD ~ 1/f ^α, α = 0)¹⁹. This treatment may be impractical on the nanoscale, because nanoscale phenomena typically differ from those on the macroscopic and mesoscopic scales. Gong et al. and Aluru et al. discovered the unidirectional transportation of water molecules across a water nanochannel in the presence of an asymmetric potential, in 2007 and 2008, respectively^20,21. In 2012, Detcheverry et al. demonstrated that the center of mass for a confined fluid performs a non-Markovian random walk in nanoporous structures, which is inconsistent with its behavior in macrosystems²². Initially, the existence of unidirectional transportation in nanosystems was controversial^23-25. However, Wan et al. proved that the unidirectional transportation of molecules at the nanoscale was feasible in the presence of an appropriate asymmetric potential and a finite autocorrelation time of thermal noise²⁶. They further suggested that thermal noise is crucial to unidirectional transportation in asymmetric nanosystems, and that in water it has an autocorrelation time of the order of picoseconds²⁶. In other words, thermal noise in water can generate molecular driving forces in thermally fluctuating conditions. Previous theoretical and experimental studieshave also shown that thermal noise cannot be treated simply as white noise on the molecular scale, it may be colored (PSD ~ 1/f ^α, α ≠ 0)^27-29. In 2014, using simulations modeling different thermostat coupling methods and parameters, we discovered that the autocorrelation time of thermal noise is intrinsic. This means it depends only on the system temperature, and does not need to be evaluated through calculation, with a duration of ~10 ps in water²⁷. In 2016, we proposed (using power spectrum analysis) that the thermal noise of polar solute molecules (such as water, methanol, etc.) in water resembles a 1/f noise distribution (a typical non-white noise) on the nanoscale²⁸. In 2019, Mukherjee et al. further confirmed that the thermal noise of protein, which is sensitive to energy fluctuations, behaves in water as 1/f noise²⁹. Non-white thermal noise appears to be crucial to understanding the structural and dynamical characteristics of molecules or polymers in water. However, our previous work only suggested that the picosecond autocorrelation time of thermal noise may be due to the alteration of the water’s hydrogen bond network²⁷. Detailed research should clarify the cause of the picosecond autocorrelation time of thermal noise.

Rotation is the most basic form of molecular motion. The rotation of water molecules in the hydration layer of solutes influences various biochemical processes, including ion transport, protein folding, and chemical reactions in the solution. For example, by calculating molecular dynamics for a model Cl⁻ + CH₃ClS_N2 reaction in water, Gertner et al. observed that rotations induced the rearrangement of water molecules’ hydrogen bonds and generated the conditions required for reactants to enter a transition state³⁰. Bernal et al. proposed that the rotations of water molecules determine proton transport in water, they found that the motion (including rotation) of neighboring water molecules was a key feature in the necessary solvent reorganization³¹. Our earlier research (2018) also demonstrated that limiting the rotations of the outermost water molecules can modulate the evaporation of water on a uniformly and completely wetted surface³².

In this study, we initially employ a molecular dynamic simulation^33-39 to compare the autocorrelation behavior of thermal noise in bulk water with those of the hydrogen bonds and molecular rotations of water molecules. We discover that the molecular rotations of water molecules have an autocorrelation time of ~10 ps, which is equal to the time of thermal noise autocorrelation in water. Moreover, the rotational autocorrelation function of water molecules closely resembles that of their thermal noise and hydrogen bonds. The variational trends in characteristic time for the autocorrelation functions of molecular rotations, hydrogen bonds, and thermal noise in water correspond to the changes in environment temperature are the same. We then study the effects of molecular rotation on the hydrogen bond network. In further simulations, we partly constrain the rotations of water molecules, by applying a uniform electric field of varying strengths. The simulation results show that restricting the rotations of water molecules can increase the lifetime of hydrogen bonds in water, indicating that molecular rotation determines the stability of the hydrogen bond network. The rotations of water molecules can alter the water’s hydrogen bond network; thus, the picosecond molecular relaxation time determines the finite lifetime of hydrogen bonds in water. In addition, the intrinsic ~10 ps autocorrelation time of thermal noise is thought to originate from alterations to the water’s hydrogen bond network²⁷. Therefore, the detailed cause of the picosecond autocorrelation time of thermal noise can be attributed to the finite rotation relaxation of water molecules in bulk water, where the hydrogen bonds appear to function as a bridge. Moreover, we find that our calculation method (choosing the oxygen atom of water to consider the autocorrelation characteristics of thermal noise) describes the rotational behavior information of molecules. As predicted, when the rotational freedom of the single-file water chain is restricted in a nanotube or nanopore, the thermal noise has a longer autocorrelation time and unidirectional transportation becomes significant^20,21,40,41. Finally, we analyze the thermal noise power spectrum in a broad frequency domain for water molecules and found that thermal noise can be modeled as white noise below the terahertz frequency domain (<0.1 THz); however, it must be regarded as non-white noise in the terahertz and higher frequency domains. The dynamic nanoscale processes usually occur on picosecond timescales (corresponding to the terahertz band). Our findings here are significant for the understanding of anomalous nanoscale behaviors, and have implications for the application of terahertz technologies in observing and detecting the structural and dynamical information of molecules in solutions.

Methods

As shown in Figure 1, we placed water molecules into a simulation box (dimensions L_X = L_Y = L_Z = 3.0 nm) then began the simulations. All simulations were performed with a canonical ensemble using the classical molecular dynamics programming package GROMACS (version 4.5.5)⁴². The simulation system used 895 TIP3P rigid water molecules⁴³ because the TIP3P water model is widely used for studying the liquidity and transport properties of water. The interaction parameters of the water molecules were modeled with an OPLS/AA force field. The atomic charges of oxygen and hydrogen were −0.8476 e and 0.4238 e, respectively. The Van der Waals radius (σ_O) and well depth (ε_o) for an oxygen atom were 0.316557 nm and 0.605194 kJ/mol, respectively; the Van der Waals radius and well depth for a hydrogen atom were 0 nm and 0 kJ/mol, respectively. Under the OPLS/AA force field, the Lennard–Jones parameters between different atoms were calculated using the geometric average for σ_ij=(σiσj)^1/2 and εij=(εiεj)^1/2; thus, the non-bonded interaction $E_{nb}={\displaystyle \sum }_i{\displaystyle \sum }_j\left[q_i{q}_j{e}^2/r_{ij}+4{\epsilon }_{ij}\left({\sigma }_{ij}^{12}/r_{ij}^{12}-{\sigma }_{ij}^6/r_{ij}^6\right)\right]$. The bond length between a hydrogen and oxygen atom in a water molecule was 0.09572 nm. The bond angle of a water molecule (denoted as H−O−H) was 104.52º. Periodic boundary conditions were applied in all directions. Nosé–Hoover thermostats^44,45 were used to maintain the system at a set of reference temperatures, which in this study were 285, 300, 315, 330, 345, and 360 K. Nosé–Hoover thermostats depend on two significant coupling parameters: the time constant for coupling ${\tau }_T$ and the period for coupling $t_{\text{couple}}$ for a particle are replaced by $\frac{{\text{d}}^2{r}_i}{\text{d}{t}^2}=\frac{F_i}{m_i}-\zeta \frac{\text{d}{r}_i}{\text{d}t}$. Here, r_i is the coordinate of the i-th atom, F_i is the force exerted on the i-th atom, and m_i denotes the mass of the i-th atom. The heat-bath parameter ζ can be expressed as $\frac{\text{d}\zeta }{\text{d}t}=\frac{4{\pi }^2}{{\tau }_{\text{T}}^2{T}_0}\left(T-T_0\right)$, where ${\tau }_{\text{T}}$ is the coupling time constant, which determines the relaxation time with which the system changes from the instantaneous temperature T to the reference temperature $T_0$. The coupling period $t_{\text{couple}}$ is the time interval between two applications of the temperature-coupling method. Under the Nosé–Hoover approach, these two coupling parameters cause an oscillatory relaxation in the system, resulting in a correct canonical ensemble. The particle mesh Ewald⁴⁶ integration was used to treat the long-range electrostatic interactions. The simulation time for each system was set as 20 ns with a time step of 2 fs. The trajectory was measured every 0.2 ps (simulation results using different recording-time resolutions can be found in the ESI†), and data from the previous 10 ns were extracted to analyze the characteristics of thermal noise.

Figure 1.Full-screen View

(Color online) The initial simulation state with 895 TIP3P water molecules in a 3.0 × 3.0 × 3.0 nm³ cube (simulation results with different box sizes can be seen in the ESI†). The red and gray spheres represent the oxygen and hydrogen atoms of the water molecules, respectively. The blue lines represent the boundaries of the periodic box in the simulation system.

Results and Discussion

Our simulations (Figure 2) show that the dynamical behaviors (including the thermal noise, hydrogen bonds, and rotations) of water molecules in bulk water have an intrinsic picosecond autocorrelation time, and that the autocorrelation time of thermal noise is closely related to the autocorrelation characteristics of the hydrogen bonds and rotations of the water molecules. Typically, thermal noise, which is also referred to as thermal agitation, is caused by the random motions of particles such as electrons and ions. In molecular dynamics simulations, the thermal noise can be characterized as the randomly fluctuating force that a molecule or atom receives from its surrounding environment. Thus, we used the fluctuating force (f) exerted on the oxygen atoms of water by surrounding water molecules to represent thermal noise in the simulations. The autocorrelation function of thermal noise is

Figure 2.Full-screen View

(Color online) (a) The autocorrelation functions $C\left(t\right)$ for different quantities of water molecules in bulk water. The solid black, solid blue, and dotted red curves represent the autocorrelation functions of the thermal noise ($C_{\text{noise}}\left(t\right)$), hydrogen bonds ($C_{\text{H}-\text{bond}}\left(t\right)$), and molecular rotations ($C_{\text{orientation}}\left(t\right)$) of water molecules in bulk water, respectively. The inset displays the function using the expression of $W\left(t\right)=t·C\left(t\right)$, in which the autocorrelation time $t$ (with respect to the maximum value of $W\left(t\right)$) is the characteristic time of the autocorrelation function. The solid black, solid blue, and dotted red curves represent the functions for the thermal noise, hydrogen bonds, and molecular rotations, respectively. (b) The characteristic time of the autocorrelation function with different temperatures (T), which were fitted using the autocorrelation function described by a single exponential function $y=e^{\frac{-t}{\tau }}$. The solid black, blue, and red curves represent the characteristic times of the thermal noise ${\tau }_{\text{noise}}$, hydrogen bonds ${\tau }_{\text{H}-\text{bond}}$ (also referred to as the lifetime of the hydrogen bond), and water molecular rotation ${\tau }_{\text{rotation}}$ in water at different temperatures, respectively. The error bars of characteristic times for thermal noise, hydrogen bonds, and molecular rotations (with respect to the temperature (285 K ≤ T ≤ 360 K)) did not exceed 0.02 ps.

Here, f(0) and f(t) represent the forces exerted on the oxygen atom of a water molecule at times 0 and t, respectively. We first calculated the autocorrelation of thermal noise in bulk water at the reference temperature of 300 K. It can be seen from Fig. 2(a) that C_noise(t) decays exponentially from 1 to 0 as the autocorrelation time t is increased. The maximum autocorrelation time (the minimum time in which C_noise(t)≈0) of thermal noise in bulk water was approximately 10 ps. In our previous work, we speculated that the finite correlation time of thermal noise in water was related to the dynamical behavior of the water’s hydrogen bond network²⁷, because water molecules in bulk water maintain their tetrahedral coordination shells by forming hydrogen bonds with other water molecules in the vicinity. Hydrogen bonding is not only the main component of the interaction between a water molecule and its neighboring water molecules, it also strongly influences the physical and chemical properties of water^47-50. Hydrogen bonds are likely to form between neighboring water molecules if the O-H bond of one molecule faces the oxygen of the next. In this study, we adopted a commonly used geometric definition of hydrogen bonding, to determine whether or not a hydrogen bond is formed. Specifically, two water molecules were considered to be hydrogen-bonded if the distance between the oxygen atoms of the molecules (O…O) was less than 3.5 Å and, at the same time, the angle between the H-O bond of the water molecule and the connecting line of two oxygen atoms (H―O…O) was less than 30° ^51-53. The autocorrelation function expression of a hydrogen bond is

where h(0) = 1 indicates that two water molecules initially form a hydrogen bond, and h(t) = 1 indicates that the these two water molecules are still bonded at time t; otherwise, h(t) = 0. C_H-bond(t) is indeed similar to C_noise(t), resulting in a comparable autocorrelation time (~ 10 ps) (see also Figure 2(a)). It is worth noting that the intrinsic dynamics of water molecules—for instance, rotations and translations—profoundly affect the lifetime of their hydrogen bonds. Thus, we further studied the autocorrelation of rotations for molecules in bulk water. A water molecule can be modeled as a set of three points, corresponding to a negatively polarized oxygen O and two positively polarized hydrogen H₁ and H₂. The angle between, and the respective lengths of, the vectors $\overrightarrow{{\text{OH}}_1}$ and $\overrightarrow{{\text{OH}}_2}$ can fluctuate. The water dipole moment vector and the vector identified from the two hydrogen atoms of each water molecule are considered here. Such molecular orientations can be conveniently described with a choice of vectors, defined as $\overrightarrow{d}=\frac{\overrightarrow{{\text{OH}}_1}+\overrightarrow{{\text{OH}}_2}}{\left|\overrightarrow{{\text{OH}}_1}+\overrightarrow{{\text{OH}}_2}\right|}$. Likewise, the autocorrelation function of molecular orientation is

where d(0) and d(t) are the dipoles of tagged water molecules at times 0 and t, respectively. As shown in Figure 2(a), it can be seen that the autocorrelation function of thermal noise for water molecules is almost identical with that of water’s molecular orientation. To facilitate a quantitative comparison, we used the single exponential functions $y\left(t\right)={\text{e}}^{\frac{-t}{{\tau }_{noise}}}$, $y\left(t\right)={\text{e}}^{\frac{-t}{{\tau }_{H-bond}}}$, and $y\left(t\right)={\text{e}}^{\frac{-t}{{\tau }_{orientation}}}$ to fit the autocorrelation functions of the thermal noise, the hydrogen bonds, and the orientations of the water molecules, respectively. Here, ${\tau }_{\text{noise} }$ is the characteristic time of the thermal noise autocorrelation, ${\tau }_{\text{H}-\text{bond} }$ is the characteristic time of the water hydrogen bond autocorrelation (also referred to as the lifetime of the hydrogen bond), and ${\tau }_{\text{orientation} }$ is the characteristic time of the molecular orientation autocorrelation. The values were 1.90, 1.38, and 2.10 ps for the ${\tau }_{\text{noise}}$, ${\tau }_{\text{H}-\text{bond} }$, and ${\tau }_{\text{orientation}}$ of water molecules in bulk water at 300 K, respectively. It is worth noting that the characteristic time of hydrogen bond autocorrelation was smaller than that of thermal noise or water molecule rotation; this was a result of the hydrogen bond definition (i.e., the O–O distance ≤ 3.5 $$ Å and hydrogen-bond angle ≤ 30°)^51-53. Because the interaction between a hydrogen donor and oxygen acceptor remains after breaking of the hydrogen bond, the average lifetime of the hydrogen bond should be smaller than the characteristic time for thermal noise or water molecule rotation. Furthermore, we used $W\left(t\right)=t·C\left(t\right)$ to model the characteristic time $\tau$, because $W\left(t\right)$ has a maximum when $t=\tau$. The inset of Figure 2(a) illustrates that the value of t (with respect to the maximum positions of $W\left(t\right)$) for the thermal noise, hydrogen bonds, and water orientations are almost identical, which further confirms that their respective characteristic times are also almost identical.

The thermal kinetic behavior of molecules is determined by temperature, because the reference temperature of the system determines the velocity distribution of the water molecules undergoing Brownian motion. Molecular velocity affects the collisions between the water molecules and their neighbors. At a certain temperature, the duration of the interaction between neighboring molecules is intrinsic. Therefore, the thermal noise correlation time of water molecules may also depend on the temperature, if it is also intrinsic. Thus, we proceeded to study the effect of bulk water temperature on the characteristic times of the thermal noise ${\tau }_{\text{noise}}$, the hydrogen bonds ${\tau }_{\text{H}-\text{bond}}$, and the molecular rotations ${\tau }_{\text{rotation}}$. It can be seen from Figure 2(b) that ${\tau }_{\text{noise}}$ decreases significantly under an increase in temperature, which confirms that the autocorrelation time of thermal noise in bulk water depends on the environment temperature. Interestingly, ${\tau }_{\text{rotation}}$ and ${\tau }_{\text{H}-\text{bond}}$ also decrease as the temperature increases. This phenomenon may be attributable to fact that an increase in the reference temperature can accelerate the molecular rotational velocity of water, which causes the finite relaxation time of molecular rotation to decease. Meanwhile, the molecular rotation can affect the neighboring hydrogen bond network; thus, the hydrogen bond lifetime should decrease as the temperature increases. Notably, the changing trends of ${\tau }_{\text{noise}}$ with respect to temperature are highly consistent. These facts indicate that the cause of the intrinsic picosecond autocorrelation time of thermal noise in water is the finite molecular rotation relaxation, in which the hydrogen bonds act as a bridge.

The hydrogen bond is the bridge between molecular rotation and thermal noise, which suggests that limiting the molecular rotation will prolong the dynamic adjustment time of the hydrogen bond network. To study the effects of molecular rotation on the alteration of the neighboring hydrogen bond network, we partly constrain the rotations of water molecules by applying a uniform electric field. Thus, we conducted several molecular dynamics simulations with an electric field intensity of 1, 2, 3, 4, and 5 V/nm. As shown in Fig. 3, the value of ${\tau }_{\text{H}-\text{bond}}$ is 1.38 ± 0.01 ps in the absence of an external electric field (i.e., E = 0 V/nm). In contrast, the lifetime of the hydrogen bonds in water increases with the electric field increase, its value is 1.39 ± 0.02 ps, 1.43 ± 0.02 ps, 1.48 ± 0.02 ps, 1.61 ± 0.02 ps, and 1.78 ± 0.03 ps for E = 1, 2, 3, 4, and 5 V/nm in the presence of an external electric field, respectively. This confirms that the lifetime of a hydrogen bond increases with the constraint strength under an increase of molecular rotation. Because the water molecule dipole is constrained by the electric field, the molecular rotation relaxation time increases as the constraint strength of molecular rotation increases. Combined with the fact that the intrinsic picosecond autocorrelation time of thermal noise stems directly from the alteration of the hydrogen bond network²⁷, we can conclude that the origin of the intrinsic picosecond autocorrelation time of thermal noise is the finite relaxation time of molecular rotations in water.

Figure 3.Full-screen View

The black squares with error bars represent the lifetimes of hydrogen bonds formed by water molecules in bulk water when applying different uniform electric fields to partly constrain the molecular rotation.

To confirm the relationship between the water molecule rotations and the thermal noise further, we analyzed the dynamic trajectories of water molecules in bulk water and calculated the angle θ between the molecular dipole and thermal noise vectors. Figure 4(a) shows the angle between the thermal noise and water molecule dipole vectors with respect to the simulation time t, where we rotated the water dipole vector in a clockwise direction until it aligned with the thermal noise vector. The angle can thus be changed from 0º to 360º. From the statistics of the time-dependent angle in Fig. 4(a), we found that the angle θ between the molecular dipole vector and the noise vector predominantly fluctuated around 180º, as indicated by the red line in Fig. 4(a). Our subsequent calculations (Fig. 4(b)), based on the probability distribution of the angle θ between the molecular dipole and noise vectors, show two peaks located at 156.9º and 213.1º. This implies that the orientation of thermal noise is almost identical with that of the molecular dipole. In our previous work²⁷, we speculated that the intrinsic picosecond autocorrelation time of thermal noise in water is directly derivable from the hydrogen bond network alteration. Moreover, the molecular rotation of water can induce the generation and fracture of hydrogen bonds and the water dipole vector is aligned with the thermal noise vector. These facts further confirm that the thermal noise of water molecules depends on the water molecule’s rotations.

Figure 4.Full-screen View

The change and distribution of the angle θ between the molecular dipole and noise vectors, in the clockwise direction; here, we used the molecular dipole as the initial vector. (a) The angle θ between the molecular dipole and noise vectors with respect to the simulation time t. The dashed gray lines represent the upper and lower limits of error for the value of the time-dependent angles. (b) The probability distribution of the angle θ between the molecular dipole and noise vectors. Angles are measured in degrees.

Thermal noise is a vector, which implies that the autocorrelation of thermal noise contains magnitude and direction information. Because the autocorrelation function is normalized, the magnitude of autocorrelation is averaged, thus the orientation plays a crucial role. However, the thermal noise vector is aligned with that of molecular rotation, which entails that the thermal noise and water dipole have near-identical autocorrelation functions. This means that our method of computing the fluctuating force on oxygen atoms in water^26-28 contains information pertaining to the water’s dynamical behavior. Specifically, it characterizes the autocorrelation behavior of the water dipole. It is worth noting that molecules are treated as spherical particles in the traditional theory of Brownian motion; thus, the traditional theory only treats the randomly fluctuating force (that of the surrounding environment on the particle) as thermal noise. As a result, it can only represent translational information of the dynamical molecular behavior. However, the structure of a molecule is directly related to its rotation, and the rotations of the molecule directly affects its dynamic properties. Therefore, molecules cannot be simply treated as spherical particles on the nanoscale, and considering the rotations of molecules can better explain the dynamical behavior of molecules at these scales.

In traditional theory, the thermal fluctuation (thermal noise) is treated or modeled as white noise. White noise is a specific type of noise that describes a random signal or process whose autocorrelation time is zero and whose power spectral density is constant. We have detailed the cause of the intrinsic thermal noise autocorrelation time in bulk water and know that it is on the picosecond scale, indicating that thermal noise is not white noise on the nanoscale. Here, we further study the non-white behavior of thermal noise. The power spectrum function is widely used to analyze fluctuation properties in a random process⁵⁴; it describes the distribution of noise power with respect to frequency, and with this we can indicate the periodicities that dominate the whole process of fluctuation and distinguish the colors of the noise^55-57. The spectral power of stationary stochastic processes is the Fourier transform of the autocorrelation function. Because the autocorrelation function of thermal noise C_noise(t) is time-reversible (in other words, C_noise(t) is an even function), the power spectrum of thermal noise can be expressed as

Here, t_m is the maximum time considered for C_noise(t), which was here set as 1 ns. Figure 5(a) shows the autocorrelation function of thermal noise in water, the thermal noise autocorrelation time is ~ 10 ps. We used the exponential function R(t)=e^-t/1.90 to fit the autocorrelation function of thermal noise. It can further be seen from Fig. 5(a) that R(t) fits the autocorrelation function of thermal noise well, which means that the characteristic time of thermal noise is 1.90 ps. Figure 5(b) shows the spectral density of the thermal noise power spectrum, which was derived from the data in Fig. 5(a) using Eq.(4). The spectral density is constant when the frequency ω is less than 1 × 10¹¹ Hz, it then decreases as frequency increases from 1 × 10¹¹ Hz to 2.5 × 10¹² Hz. Because the Fourier transform of y(t)=e-at is $S\left(\omega \right)=\frac{2a}{a^2+{\omega }^2}$, S(ω) should describe the power spectrum of thermal noise well. From Fig. 5(b), it can be seen that the spectral power of thermal noise indeed approximates to the function of $S\left(\omega \right)=\frac{2a}{a^2+{\omega }^2}$, where $a=\frac{1}{{\tau }_{noise}}$ = 5.26 × 10¹¹ with the frequency domain ω ranging from 1 × 10⁹ Hz to 2.5 × 10¹² Hz (see the inset of Figure 5(b) for the spectral density of thermal noise in a wider frequency range, obtained from $S\left(\omega \right)=\frac{2a}{a^2+{\omega }^2}$). According to the expression for S(ω), we know that S(ω) approximates to a constant of 2/a when ω< 0.1 THz, because ω is considerably smaller than a in this range. When ω > 0.1 THz, ω is comparable to a, and S(ω) is no longer constant. This suggests that thermal noise in bulk water takes the form of white noise in the lower frequency domain and non-white noise at terahertz and higher frequency domains (see the ESI†for further discussion). Because the particle dynamics we observed on macroscopic or mesoscopic scales are low frequency, it may be feasible to treat the thermal noise as white noise on these scales. This treatment may not be appropriate at the nanoscale, because the energy of a nanoscale system is expressed by several k_BT ranges (k_B represents the Boltzmann constant); thus, the magnitude of the thermal fluctuations is comparable to the thermodynamic quantities to which the system is subjected. The molecular or atomic collisions occur within a timescale of picoseconds or faster; this corresponds to high-frequency dynamics. Consequently, thermal noise is non-white at nanoscales. Thus, the biased motion of particles resulting from thermal noise may be possible in nanoscale systems if (a) spatial inversion asymmetry holds⁵⁸ and (b) we can prevent distorted measurement in high time-resolution experimental techniques by distinguishing the noise color¹⁸.

Figure 5.Full-screen View

(Color online) The autocorrelation function of thermal noise in the time and frequency domains, respectively. (a) The solid black curve represents the autocorrelation function of thermal noise within time t from 0 ps to 40 ps. The dashed red curve represents the fitting function $R\left(t\right)=e^{\frac{-t}{{\tau }_{noise}}}$ of the thermal noise autocorrelation function, where ${\tau }_{\text{noise}}$ = 1.90 ps is the characteristic time of thermal noise. (b) The solid black curve represents the spectral density of thermal noise in bulk water with respect to the frequency $\omega$, which was calculated from the thermal noise autocorrelation function within the frequency domain from 1×10⁹ Hz to 2.5×10¹² Hz. The red curve is the fitting function of $S\left(\omega \right)=\frac{2a}{a^2+{\omega }^2}$, where $a=\frac{1}{1.90\times {10}^{-12}}=$ 5.26 × 10¹¹; the inset illustrates the fitting function of S(ω)with a wider frequency domain (from 1 × 10³ Hz to 1 × 10¹⁶ Hz).

Conclusion

We used molecular dynamic simulations to investigate the relationship between the autocorrelation behaviors of thermal noise, hydrogen bonds, and the molecular rotations of water molecules in bulk water. The simulation results showed that the thermal noise autocorrelation function of water molecules in bulk water is very similar to the autocorrelation functions of the hydrogen bonds and molecular rotations of water molecules. Moreover, the characteristic autocorrelation times of the thermal noise, hydrogen bonds, and water rotations exhibited the same behaviors under changes in environment temperature. Further study on the thermal noise power spectrum of water molecules shows that thermal noise on the nanoscale cannot be regarded as white noise. We found that the cause of the intrinsic picosecond autocorrelation time for thermal noise is the finite (picosecond) molecular rotation relaxation in water. The underlying mechanism arises because the rotation of water molecules induces the alteration of their hydrogen bond network, indicating that the finite molecular rotation relaxation is the origin of the finite autocorrelation time of hydrogen bonds in water. Meanwhile, the thermal fluctuating force occurs from the alteration of the hydrogen bond network in water²⁷, meaning the hydrogen bond plays the role of a bridge. Moreover, we found that the orientation of the molecular dipole is nearly opposite that of the thermal fluctuating force exerted on the oxygen atom of the water molecule. This result indicates that our calculation method for the thermal noise, that of choosing the oxygen atom of the water molecule to consider the autocorrelation time of thermal noise in water, provides additional information about the molecular rotation behavior.

The finite water molecular rotation relaxation time causes the finite autocorrelation time of thermal noise in water. Therefore, the thermal noise under confined conditions (such as a water channel of nanometer size) might have a longer autocorrelation time when the rotational freedom is restricted. Therefore, the unique dynamical behavior of a single-file water chain confined in a nanotube or other porous material of similar size, in the presence of an appropriate asymmetric potential, may be made more understandable^20,21. It should also be noted that many physical and biochemical processes occur on the picosecond timescale, and that this timescale corresponds to the terahertz ⁵⁸ frequency domain. Accordingly, our findings are expected to further understanding by using terahertz technology to observe and detect the unique physical and biochemical processes occurring on the nanoscale, and to promote the applications of terahertz technology in biomedical science.