Question

6.12 Simple estimate of first passage times The rate of a molecule passing over a free-energy barrier of height $B$ is proportional to the Boltzmann factor $e^{-B/k_{B}T}$ with $T$ the (absolute) temperature and $k_{\mathrm{B}}$ the Boltzmann constant. The exponential dependence on $B$ can be understood heuristically as follows: Thermal fluctuations typically give the molecule an energy of about $k_{\mathrm{B}}T.$ To go up in energy by another factor of $k_{\mathrm{B}}T$ is somewhat more unlikely than to go down. Consider the barrier of height $B$ as being a staircase of energy steps each of height $k_{\mathrm{B}}T.$ Next suppose that the probability that the next step of the particle is up is $q$ and the probability that the next step is down is given by $1-q$ with $q<1/2$. Make a crude estimate of the probability $p$ of getting to the top of the barrier in "one try." You will need to say what you mean by "one try." Compare your answer with the Boltzmann factor for $B\gg k_{\mathrm{B}}T.$ (Problem courtesy of Daniel Fisher.)

Short answer

The probability of a molecule overcoming a free energy barrier in 'one try', defined as a single series of attempts to ascend the barrier without falling, is given by $p=(\frac{e^{-1}}{(1+e^{-1})}{)}^{B/(k_{B}T)}$. This aligns with the Boltzmann factor when $B\gg k_{B}T$, indicating the molecular inclination to occupy lower energy states.

Step-by-step solution

Define the concept of a 'try'

For the purpose of this exercise, it can be stated that a 'try' is one full attempt to ascend the staircase or energy barrier, starting from the base and either reaching the top or falling back down.

Establish the nature of the barrier and 'travelling' over it

The barrier is comprised of $\frac{B}{k_{B}T}$ steps, each of height $k_{B}T$. Ascending each step represents a rise in energy, which is counteracted by a thermal fluctuation that pushes the molecule back down.

Calculate the probability of ascending each step

The ratio of probabilities of going up versus going down for each step equals the Boltzmann factor, $e^{-(k_{B}T)/(k_{B}T)}=e^{-1}$. The probability $q$ of advancing up one step is the Boltzmann factor divided by the sum of all possible outcomes (going up and down), which is $q=\frac{e^{-1}}{1+e^{-1}}$.

Estimate the probability of reaching the top in 'one try'

The probability $p$ of successfully ascending all steps without falling is equivalent to the product of probabilities of ascending each individual step: $p=q^{B/(k_{B}T)}=(\frac{e^{-1}}{(1+e^{-1})}{)}^{B/(k_{B}T)}$.

Compare with the Boltzmann factor

When $B\gg k_{B}T$, the probability $p$ approximates the Boltzmann factor for a barrier height $B$, and demonstrates the thermodynamic preference for lower energy states, as expected from the Boltzmann distribution.

Key concepts

Free-energy Barrier

In the field of thermodynamics and statistical mechanics, a free-energy barrier represents an obstacle that parties, like molecules or particles, must 'overcome' to undergo a transformation or a reaction. Imagine you're hiking and encounter a hill: to get to the other side, you must first climb the hill. In molecular terms, the 'hill' is the energy that the system must gain to transition between different states.

Much like our hiker who needs extra energy to climb, a particle must gain energy to reach the top of the free-energy barrier. This energy is often provided by thermal fluctuations, random energy changes due to the heat within the system. The height of this barrier, denoted by the letter $B$, is crucial because it determines the rate at which the transformation occurs. The taller the barrier, the less likely the particle is to cross it spontaneously - you would need more energy to climb a taller hill, after all.

Boltzmann Factor

The Boltzmann factor, symbolized by $e^{-\frac{B}{k_{B}T}}$, is a key concept in the realm of physics and chemistry as it provides a way to calculate the likelihood of a particle overcoming a free-energy barrier at a given temperature $T$. The factor's name celebrates Ludwig Boltzmann, a pioneer in statistical mechanics.

The term $k_B$, known as the Boltzmann constant, and temperature $T$ in the exponent's denominator play central roles. They dictate the scale of the thermal energy available in the system. Higher temperatures or smaller barriers make the Boltzmann factor larger, which in turn increases the probability that a particle will have enough thermal energy to cross the barrier without external assistance. It's a bit like increasing the chance of our hiker making it over the hill by lowering the hill's height or giving them more energy.

Thermal Fluctuations

Imagine you're sitting by a calm lake, and all of a sudden, a stone is thrown into it, creating ripples: that's akin to thermal fluctuations. In the microscopic world, molecules in a thermodynamic system experience constant random motion, due to the energy transferred from the heat of their environment. These random changes in energy levels—fluctuations—happen regardless of whether the system as a whole is in equilibrium or not.

These fluctuations can briefly provide a molecule with enough energy to cross a free-energy barrier. In the context of the exercise, a molecule's typical energy provided by thermal fluctuations is about $k_{B}T$, just enough to nudge our molecular 'hiker' into deciding to take that step up the energy staircase.

Probability Estimation

Probability estimation is the process of quantifying the likelihood of an event occurring. In our molecular hike over an energy barrier, we use probabilities to estimate the chances of a particle climbing to the top in a single 'try'. We define 'one try' as an individual instance in which a particle makes an independent attempt to surmount the barrier, without external influences.

In the context of our energy staircase, the calculation hinges on the Boltzmann factor. By treating each step up the barrier as an independent event, and considering the probability $q$ of moving up versus down ($1-q$), we can estimate the total probability $p$ by raising the chance of climbing one step to the power corresponding to the number of steps in the barrier. The calculation shows that as the barrier becomes significantly larger than the thermal energy ($B\text{(}\gg$ k_{B} T\)), the probability of crossing it without external influences drops sharply, emphasizing the power of the Boltzmann factor in determining molecular behavior.